PRINCIPLES 

OF 


ELECTRICAL 
ENGINEERING 


BY 

HAROLD  FENDER,  PH.D. 

/  ? 

PROFESSOR  OF  THEORETICAL  AND  APPLIED  ELECTRICITY 

MASSACHUSETTS  INSTITUTE  OF  TECHNOLOGY 

MEM.  A.  I.  E.  E. 


FIRST  EDITION 


McGRAW-HILL   BOOK   COMPANY 
239  WEST  39TH  STREET,  NEW  YORK 

6  BOUVERIE  STREET,  LONDON,  E.  C. 
1911 


COPYHIGHT,    1910,    1911 
BY    THE 

McGRAW-HiLE»  BOOK  COMPANY 


The  Plimpton  Press,  Norwood,  Mass.,  U.S.A. 


PREFACE 

THIS  book  is  an  exposition  of  the  physical  principles  upon 
which  the  art  of  electrical  engineering -is  based,  together  with  a 
discussion  of  the  application  of  these  principles  in  some  of  the 
simpler  forms  of  electric  apparatus  and  machinery.  The  first 
four  chapters  of  the  book  may  be  looked  upon  as  an  introduction 
to  the  detailed  study  of  continuous  current  machinery,  the  last 
five  to  the  detailed  study  of  alternating  current  machinery  and 
the  transmission  and  distribution  of  energy  by  alternating  cur- 
rents. The  text  is  the  development  of  a  course  of  lectures 
given  by  the  author  to  the  junior  class  in  electrical  engineer- 
ing at  the  Massachusetts  Institute  of  Technology.  The  time 
allotted  to  this  course,  exclusive  of  home  preparation,  is  fifty- 
five  hours  of  lectures,  twenty-three  hours  of  recitations,  and 
twenty-three  hours  for  the  solution  of  problems  under  the 
supervisi6n  of  an  instructor. 

A  preliminary  edition  of  the  book  was  published  in  1910, 
primarily  for  the  use  of  the  students  at  the  Institute,  and  this 
preliminary  edition  was  preceded  by  mimeographed  notes  cover- 
ing practically  the  same  ground.  The  present  edition  differs 
from  the  preliminary  edition  chiefly  in  the  addition,  at  the  end 
of  each  chapter,  of  a  summary  of  the  important  definitions  and 
principles  developed  in  that  chapter  and  of  a  list  of  problems, 
with  answers,  illustrative  of  these  principles.  Certain  sections  of 
the  text  have  also  been  rewritten,  particularly  the  sections  on 
energy,  inductance,  and  capacity;  the  conception  of  linkages 
between  the  electric  and  magnetic  circuits  has  also  been  more 
fully  developed.  The  typographical  errors  of  the  preliminary 
edition  have  been  corrected  and  additional  steps  have  been 
inserted  in  some  of  the  mathematical  deductions. 

There  will  undoubtedly  be  those  who,  after  an  examination 
of  the  text,  will  pronounce  it  more  of  a  treatise  on  physics  than 
on  electrical  engineering.  But  electrical  engineering  is  primarily 
the  application  of  physics,  and  of  necessity  the  principles  of 
electrical  engineering  are  the  principles  of  physics.  This  fact  is 


235471 


vi  PREFACE 

too  frequently  overlooked,  and  the  student  is  rushed  into  the 
study  of  electric  machinery  and  other  advanced  subjects  with 
but  the  vaguest  conception  of  the  physical  principles  upon  which 
the  operation  of  electric  apparatus ,  is  based.  In  the  author's 
opinion,  a  clear  conception  of  the  principles  of  physics  and  the 
ability  to  apply  these  principles  in  co-ordinating  the  experimental 
facts  of  physics,  both  qualitatively  and  quantitatively,  is  abso- 
lutely essential  before  one  can  get  a  clear  understanding  of  the 
more  complicated  reactions  that  take  place  in  electric  machinery 
and  transmission  circuits.  It  is  with  the  hope  that  others  may 
find  this  text  useful  in  filling  the  gap  between  elementary  physics 
and  applied  electricity  that  the  author  offers  it  to  the  public. 

The  method  of  treatment  adopted  throughout  is  to  describe 
first  certain  simple  and  typical  experiments  which  illustrate  a 
given  principle,  second,  to  state  the  principle  in  an  exact  manner 
in  its  general  form,  and  then  explain  the  application  of  the  prin- 
ciple in  one  or  more  practical  cases.  The  problems  given  at  the 
end  of  each  chapter  serve  as  a  further  illustration  of  the  principles 
developed.  The  attempt  has  also  been  made  to  make  each  sec- 
tion lead  naturally  into  the  next  and  to  show  how  the  various 
phenomena  of  electricity  and  magnetism  are  interrelated.  The 
analogy  between  the  flow  of  electricity  and  hydraulics  is  brought 
out  repeatedly,  and  emphasis  is  laid  upon  the  similarity  of 
magnetic  and  electrostatic  phenomena.  In  the  discussion  of 
alternating  currents  sine  functions  are  used  until  the  meaning/, 
of  the  various  terms,  such  as  effective  value,  phase  difference^/ 
etc.,  has  been  made  clear;  the  vector  method  is  then  introduced, 
and  finally  the  symbolic  method  is  developed. 

The  calculus  is  employed  from  the  very  beginning  of  the  book. 
The  calculus  is  taught  the  students  of  electrical  engineering  in 
practically  every  technical  school  in  the  country.  The  author 
has  adopted  the  common-sense  view  that  since  the  student  is 
provided  with  so  useful  a  tool  he  should  learn  how  to  use  it, 
particularly  as  this  tool  is  one  of  the  greatest  labor-saving  devices 
ever  invented.  In  every  case,  however,  the.  physical  meaning 
of  the  formulas  developed  has  been  clearly  stated. 

The.  problems  at  the  end  of  each  chapter  are  of  two  kinds, 
which  for  convenience  may  be  designated  as  practical  and 
theoretical.  To  fix  principles  in  the  student's  mind  problems 
may  be  devised  which,  though  seldom  met  with  in  practice,  are 


PREFACE  vil 

much  more  effective  than  purely  practical  ones.  Practical  prob- 
lems, however,  should  not  be  neglected,  for  the  student  should 
also  gain  facility  in  making  the  simple  calculations  of  ordinary 
practice.  In  addition,  many  of  the  problems  of  the  latter  class 
have  been  selected  to  bring  out  the  principles  of  certain  special 
phenomena  and  methods  of  practice  which  are  not  treated  in 
detail  in  the  text. 

To  those  who  may  use  the  book  in  the  class-room  the  fol- 
lowing suggestions  are  offered.  When  the  time  available  .is 
limited,  the  articles  printed  with  close  spacing,  for  example, 
Article  42,  may  be  omitted.  The  student  should  be  made  to 
understand  that  the  summaries  at  the  end  of  each  chapter  are 
not  to  be  memorized,  but  are  given  merely  as  a  bird's-eye  view 
of  the  chapter.  The  solution  of  as  many  problems  as  possible 
in  conjunction  with  the  study  of  the  text  will  also  enable  the 
student  to  test  his  understanding  of  the  latter;  it  is  with  this 
object  in  view  that  the  answers  to  the  problems  have  been  given. 
The  student  should  be  required  to  follow  closely  in  all  written 
work  the  recommendations  of  the  American  Institute  of  Electrical 
Engineers,  given  in  Appendix  A. 

The  author  takes  this  opportunity  of  expressing  his  indebted- 
ness to  Messrs.  Gary  T.  Hutchinson,  W.  A.  Del  Mar,  and  H.  S. 
Osborne  and  to  the  various  members  of  the  Electrical  Engineer- 
ing Department  of  the  Massachusetts  Institute  of  Technology 
for  many  valuable  suggestions  and  criticisms.  The  majority  of 
the  problems  in  this  edition  were  prepared  by  Mr.  R.  G.  Hudson, 
to  whom  the  author  is  also  indebted  for  his  assistance  in  the 
reading  of  proof. 

HAROLD  FENDER 

EAST  BLUE  HILL,  MAINE, 
August  14,  1911 


CONTENTS 

CHAPTER  I 

FUNDAMENTAL  IDEAS  AND  UNITS 

Art.  Page 

1.  Introduction 1 

2.  Length 1 

3.  Surface 2 

4.  Volume 2 

5.  Angle 3 

6.  Time 3 

7.  Displacement  .          .         .         .         .         .  .         .4 

8.  Vectors   ...........  4 

9.  Composition  of  Vectors      ........  6 

10.  Velocity .9 

11.  Acceleration     .          .         .         . 10 

12.  Mass 12 

13.  C.  G.  S.  or  absolute  System  of  Units 13 

14.  Density  and  Specific  Gravity      .......  13 

15.  Center  of  Mass 14 

16.  Linear  Momentum  and  Moment  of  Momentum     ....  15 

17.  Conservation  of  Mass,  Conservation  of  Linear  Momentum,  Con- 

servation of  Moment  of  Momentum       .....  16 

18.  Force 16 

19.  Moment  of  Force  or  Torque        .......  19 

20.  Moment  of  a  System  of  Particles  Acted  upon  by  Several  Forces        .  20 

21.  Work  and  Energy 21 

22.  Power 24 

23.  Harmonic  Motion     .........  25 

24.  Temperature    ..........  29 

25.  Heat  Energy 30 

26.  Efficiency  and  Losses          .          .          .         .         .         .         .         .31 

27.  Newton's  Law  of  Gravitation      .......  32 

Problems          .         .         .         .         .         .         .         .         .         .33 

CHAPTER  II 

MAGNETISM 

28.  Magnets.     Magnetic  Poles         .         .         .         .  „  .         .37 

29.  Paramagnetic  and  Diamagnetic  Substances  .....  37 

30.  Attraction  and  Repulsion  of  Magnetic  Poles           ....  38 

31.  Magnetic  Charge.    A  Magnetic  Pole  as  a  Force-Producing  Agent    .  38 

ix 


X  CONTENTS 

Art.  Page 

32.  Induced  Magnetisation      .          .         .         .  .         .         .39 

33.  Point-Poles      .          . 41 

34.  Proper  ties  of  Magnetic  Poles       .         .         .         .         .         .         .41 

35.  Pole  Strength  per  Unit  Area 43 

36.  Magnetic  Field  of  Force.     Field  Intensity  or  Magnetising  Force       .  43 

37.  Example  Illustrating  the  Application  of  Above  Definitions      .          .  45 

38.  Force  Required  to  Separate  Two  Equal  and  Opposite  Poles  in 

Contact 48 

39.  The  Earth's  Magnetic  Field 49 

40.  Magnetic  Moment.     Equivalent  Length  of  a  Magnet     .          .          .  49 

41.  Equality  of  the  Poles  of  a  Magnet.    Equilibrium  Position  of  a  Mag- 

net in  a  Uniform  Magnetic  Field  ......  50 

42.  Measurement  of  the  Horizontal  Component  of  the  Intensity  of  a 

Magnetic  Field  ...  .  *       .  .50 

43.  Flux  of  Magnetic  Force  Due  to  a  Single  Pole.     Lines  of  Magnetic 

Force  Due  to  a  Single  Pole   .          .          .          .          .          .          .53 

44.  Gauss's  Theorem       .........  56 

45.  Lines  of  Force  Representing  the  Resultant  Field  Due  to  any  Number 

of  Magnets         .........  57 

Intensity  of  Magnetisation          .          .         .          .          .  .62 

Lines  of  Magnetic  Induction.  Flux  of  Induction  .  .  67 

Flux  Density 71 

The  Normal  Components  of  the  Flux  Density  on  the  Two  Sides  of 

any  Surface  are  Equal          .          .                   ....  73 

50.  The  Tangential  Components  of  the  Field  Intensity  on  the  Two  Sides 

of  any  Surface  are  Equal      .......  74 

51.  Conditions  which  must  be  Satisfied  at  Every  Surface  in  a  Magnetic 

Field 75 

Induced  Magnetisation      ........  76 

Magnetic  Permeability      ....          .          .          .          .          .77 

Refraction  of  the  Lines  of  Induction  at  the  Surface  of  Separation  of 

Two  Bodies  of  Different  Permeabilities  ....  78 
Value  of  the  Pole  Strength  per  Unit  Area  Induced  on  the  Surface  of 

Separation  of  Two  Bodies  of  Different  Permeabilities        .          .  79 

56.  Field  Intensity  at  any  Point  in  a  Magnetic  Medium  of  Constant  Per- 

meability Completely  Surrounding  a  Point-Pole  and  Filling  all 

Space 80 

57.  Magnetic  Hysteresis.          .          .          .          .          .         .         .          .82 

58.  Normal  B-H  Curves.     Magnetic  Saturation          ....  86 

59.  Magnetic  Potential 88 

60.  Difference  of  Magnetic  Potential          .          .         .         .  91 

61.  Equipotential  Surf  aces       .          .          .          .          .          .  92 

Summary  of  Important  Definitions  and  Principles          ...  93 

Problems  96 


CONTENTS  xi 

CHAPTER  III 

CONTINUOUS  ELECTRIC  CURRENTS 

Art.  Page 

62.  The  Electric  Current 101 

63.  Conductors  and  Insulators  or  Dielectrics     .....  103 

64.  Electricity  Analogous  to  an  Incompressible  Fluid  Filling  all  Space  104 

65.  A  Wire  as  a  Geometrical  Line  .          .          .          .          .          .108 

66.  Definition  of  the  Strength  of  an  Electric  Current.     Definition  of  a 

Continuous  Current    ........  108 

67.  Definition  of  the  Direction  of  an  Electric  Current,  Left-Hand  Rule  111 

68.  Conductors  in  Series  and  in  Parallel    .          .          .          .          .          .112 

69.  Total  Force  Produced  by  a  Magnetic  Field  on  a  Wire  Carrying  an 

Electric  Current 113 

70.  Force  Produced  by  a  Uniform  Magnetic  Field  on  a  Straight  Wire 

Carrying  an  Electric  Current         .          .          .          .          .          .113 

71.  Magnetic  Field  Produced  by  an  Electric  Current  in  a  Wire     .          .  113 

72.  Direction  of  the  Lines  of  Force  Produced  by  an  Electric  Current     .  115 

73.  Magnetic  Field  Due  to  a  Current  in  a  Long  Straight  Wire      .          .116 

74.  Magnetic  Field  Due  to  a  Current  in  a  Circular  Coil  of  Wire    .          .118 

75.  Absolute  Measurement  of  an  Electric  Current      ....  120 

76.  Comparison  of  the  Strengths  of  Electric  Currents.     Galvonometers 

and  Ammeters   .........  122 

77.  Electrolysis  and  Electrolytes     .......  124 

78.  Electrochemical  Equivalent  of  a  Substance           ....  126 

79.  The  International  Ampere         .......  126 

80.  Quantity  of  Electricity     .         .                   127 

81.  Electric  Resistance.     Joule's  Law      ......  129 

82.  Absolute  Measurement  of  Electric  Resistance      ....  130 

83.  Specific  Resistance  or  Resistivity        ......  132 

84.  Electric  Conductance  and  Conductivity 136 

85.  Matthiessen's  Standard  of  Conductivity 136 

86.  Temperature  Coefficient  of  Electric  Resistance    .          .         .          .  137 

87.  Difference  of  Electric  Potential.     Electric  Energy         .         .          .  139 

88.  Measurement  of  Drop  of  Electric  Potential           ....  144 

89.  Ohm's  Law 146 

90.  Electric  Power  and  Electric  Energy    ......  146 

91.  ^he  Wattmeter       .          . 147 

92.  Electromotive  Force         ........  148 

93.  Generalized  Ohm's  Law 150 

94.  Contact  Electromotive  Force 153 

95.  Chemical  Batteries 155 

96.  Definition  of  the  International  Volt    ......  157 

97.  Thermal  Electromotive  Forces           ......  157 

98.  Kirchhoff 's  Laws 158 

99.  The  Wheatstone  Bridge 162 

100.  The  Potentiometer 163 

101.  Stream  Lines  of  Electric  Current  165 


xii  CONTENTS 

Art.  Page 

102.  Resistance  Drop  and  Electric  Intensity.     Electric  Equipotential 

Surfaces 167 

103.  Insulation  Resistance  of  a  Single  Conductor  Cable        .          .          .  169 

104.  Field  Intensity  at  any  Point  due  to  a  Current  in  a  Wire  of  Circular 

Cross  Section 170 

Summary  of  Important  Definitions  and  Principles         .          .          .174 

Problems         .         .         . 180 

CHAPTER  IV 

ELECTROMAGNETISM 

105.  Electromotive  Force  Due  to  Change  in  the  Number  of  Lines  of 

Induction  Linking  a  Circuit.     Linkages  between  Flux  and  Cur- 
rent              185 

106.  Work  Required  to  Change  the  Number  of  Lines  of  Induction  Link- 

ing an  Electric  Circuit  .         .         .         .         .         .          .187 

107.  Electromotive  Force  Induced  by  the  Cutting  of  Lines  of  Induction. 

Right-Hand  Rule 188 

108.  Intensity  of  the  Magnetic  Field  Inside  a  Long  Solenoid           .          .  191 
109.-    Determination  of  the  Number  of  Lines  of  Induction  Linking  an 

Electric  Circuit.     Measurement  of  Quantity  of  Electricity         .  193 

110.  Determination  of  the  B-H  Curve  and  Hysteresis  Loop  .          .          .  195 

111.  The  Continuous  Current  Dynamo      .          .          .          .          .          .197 

112.  Calculation  of  the  Electromotive  Force  Induced  in  the  Armature 

of  a  Continuous  Current  Dynamo         .          .          .          .          .  204 

113.  Magnetomotive  Force      ........  205 

114.  Magnetic  Reluctance        ........  207 

115.  Calculation  of  Ampere-Turns  Required  to  Establish  a  Given  Flux  210 

116.  Self  and  Mutual  Induction 211 

1 17.  Proof  of  the  Relation  between  Inductance  and  Linkages         .          .  213 

118.  Magnetic  Energy  of  an  Electric  Current      .....  215 

119.  Analogy  between  the  Magnetic  Energy  of  an  Electric  Current  and 

the  Kinetic  Energy  of  a  Moving  Column  of  Water  .          .218 

120.  Magnetic  Energy  of  Two  or  More  Electric  Currents  .         .          .  219 

121.  Calculation  of  Inductance.     Skin  Effect      ...         .          .  222 

122.  Self  Inductance  of  Two  Long  Parallel  Wires         .          .         .          .223 

123.  Self  Inductance  of  a  Concentrated  Winding          ....  225 

124.  Self  Inductance  of  a  Long  Solenoid 225 

125.  Total  Energy  of  a  Magnetic  Field  in  Terms  of  the  Field  Intensity 

and  Flux  Density 226 

126.  Heat  Energy  Due  to  Hysteresis 228 

126a.  Tractive  Force  of  an  Electromagnet  ......  229 

Summary  of  Important  Definitions  and  Principles         .          .          .  230 

Problems  234 


CONTENTS  xiii 

CHAPTER  V 

ELECTROSTATICS 

Art.  Page 

127.  Electric  Charges 237 

128.  Positive  and  Negative  Charges.     Attraction  and  Repulsion  of 

Charged  Bodies 238 

129.  Charging  by  Contact  and  by  Induction        .....  239 

130.  Point-Charges 242 

131.  Properties  of  Electric  Charges  . 242 

132.  Electrostatic  Field  of  Force.     Electrostatic  Intensity   .          .          .  244 

133.  Lines  of  Electrostatic  Force.     Flux  of  Electrostatic  Force      .          .  244 

134.  Lines  of  Electrisation        ........  245 

135.  Lines  of  Electrostatic  Induction.     Flux  of  Electrostatic  Induction  247 

136.  Dielectric  Constant 248 

137.  A  Closed  Hollow  Conductor  an  Electrostatic  Screen      .          .          .  249 

138.  The  Electrostatic  Intensity  Within  the  Substance  of  a  Conductor 

in  which  there  is  no  Electric  Current  is  always  Zero         .          .  249 

139.  The  Total  Charge  Within  any  Region  Completely  Enclosed  by  a 

Conductor  is  always  Zero  .......  250 

140.  The  Electrostatic  Intensity  Just  Outside  a  Charged  Conductor  is 

Normal  to  the  Surface  of  the  Conductor        ....  250 

141.  The  Electrostatic  Flux  Density  Just  Outside  a  Charged  Conductor 

is  Independent  of  the  Nature  of  the  Surrounding  Dielectric      .  251 

142.  Conditions  which  must  be  Satisfied  at  any  Surface  in  an  Electro- 

static Field .          .          .253 

143.  Electrostatic  Potential 253 

144.  Difference  of  Electrostatic  Potential  ......  254 

145.  Electrostatic  Equipotential  Surfaces  ......  255 

146.  Parallel  Plate  Electrometer 256 

147.  Relation  between  Electrostatic  Charge  and  Quantity  of  Electricity  259 

148.  Relation  between  Electrostatic  Potential  and  Electric  Potential     .  261 

149.  Displacement  Current      ........  263 

150.  Dielectric  Strength.     Discharge  from  Points.     Corona          .          .  266 

151.  Electric  Capacity.     Electric  Condenser       .....  267 

152.  Simple  Forms  of  Condensers     .......  269 

153.  Specific  Inductive  Capacity       .......  272 

154.  Condensers  in  Series  and  in  Parallel    .          .          .          .          .          .  272 

155.  Absorption  in  a  Condenser.     Residual  Charge      ....  273 

156.  Energy  of  an  Electrostatic  Field         ......  274 

157.  Dielectric  Hysteresis         ........  275 

Summary  of  Important  Definitions  and  Principles         .          .          .  276 

Problems         ...  281 


xiv  CONTENTS 

CHAPTER  VI 

VARIABLE  CURRENTS 

Art.  Page 

158.  Total  Energy  Associated  with  an  Electric  Circuit          .         .          .284 

159.  General  Equations  of  the  Simple  Electric  Circuit           .          .          .  285 

160.  Establishment  of  a  Steady  Current  in  a  Circuit  Containing  Resist- 

ance and  Inductance           .......  288 

161.  Decay    of    Current    in    a  Circuit    Containing    Resistance    and 

Inductance       .........  291 

162.  Charging  a  Condenser  through  a  Resistance         ....  292 

163.  Discharge  of  a  Condenser  through  a  Resistance    ....  294 

164.  Discharge  of  a  Condenser  through  an  Inductance          .          .          .  295 
Summary  of  Important  Relations       ......  299 

Problems 301 

CHAPTER  VII 
ALTERNATING  CURRENTS 

165.  Introduction ...  304 

166.  The  Simple  Alternator      .                                                                   .  304 

167.  Definition  of  Alternating  Current  and  Alternating  Electromotive 

Force 306 

168.  Period,  Frequency,  Alternations,  Periodicity  and  Phase         .          .  307 

169.  Difference  in  Phase           ...                   .          .                   .  308 

170.  Instantaneous,  Maximum  and  Average  Values     ....  309 

171.  Effective  Values 310 

172.  The  Use  of  Alternating  Currents        .                   ....  312 

173.  Alternating  Current  Transformer       ......  313 

174.  Average  Power  Corresponding  to  a  Harmonic  P.  D.  and  a  Har- 

monic Current.     Power  Factor  .          .          .          .          .          .314 

175.  Power  Corresponding  to  a  Non-Harmonic  P.  D.  and  a  Non-Har- 

monic Current           ........  317 

176.  Effective  Value  of  a  Non-Harmonic  Current  or  P.  D.     .          .          .  319 

177.  Equivalent  Sine-Wave  Current  and  P.  D.    .                                      .  319 

178.  Determination  of  the  Maximum  Value  and  Phase  of  the  Harmonics 

in  a  Wave  of  any  Shape      .......  320 

179.  The  Fisher-Hinnen  Method  for  Analysing  a  Non-Harmonic  Wave  .  322 

180.  Form  Factor .328 

181.  Amplitude  Factor   .  .328 

182.  Power  and  Reactive  Components  of  P.  D.  and  Current .          .          .  328 

183.  Apparent  Power      ...  .330 

184.  Reactive  Power       .                                                                             .  330 

185.  Addition  of  Alternating  Currents  and  of  Alternating  P.  D.'s             .  331 

186.  Representation  of  a  Harmonic  Function  by  a  Rotating  Vector         .  332 

187.  The  Vectors  Representing  Any  Number  of  Harmonic  Currents  and 

P.  D.'s  of  the  same  Frequency  are  Stationary  with  Respect  to 

One  Another    . 334 


CONTENTS  xv 

Art.  Page 

188.  The  Lengths  of  the  Vectors  Representing  Harmonic  Functions  Taken 

Equal  to  their  Effective  Values   ......     335 

189.  P.  D.  due  to  Harmonic  Current  in  a  Circuit  of  Constant  Resistance 

and  Inductance          .         .          .          .          .          .          .  335 

190.  Effective  Resistance,  Reactance  and  Impedance  of  an  Alternating 

Current  Circuit 337 

191.  Reactance  and  Impedance  of  a  Coil  of  Constant  Resistance  and 

Inductance  to  a  Harmonic  Current       .....     340 

192.  Reactance  and  Impedance  of  a  Coil  of  Constant  Resistance  and 

Inductance  to  a  Non-Harmonic  Current        .          .          .          .     340 

193.  Current  through  a  Condenser  when  a  Harmonic  P.  D.  is  Established 

across  it.     Capacity  Reactance  .          ...          .          .     342 

194.  Current  through  a  Condenser  when  a  Non-Harmonic  P.  D.  is 

Established  across  it  .......     343 

195.  Impedance  of  a  Resistance,  Inductance  and  Capacity  in  Series  to  a 

Harmonic  Current    ........     344 

196.  Resonance 345 

197.  Impedances  in  Series        .         .          .          .          .          .          .          .     346 

198.  Impedances  in  Parallel .    ~.     348 

199.  Conductance,  Susceptance  and  Admittance  .          .          .     349 

200.  Admittance  of  an  Inductance  and  Capacity  in  Parallel  to  a  Har- 

monic Current.     Resonance       .          .          .          .          .          .351 

201.  Transient  Effects  Produced  when  a  Harmonic  E.  M.  F.  is  Impressed 

on  a  Circuit      .........     353 

202.  Discharge  of  a  Condenser  having  Negligible  Leakance  through  a 

Resistance  and  Inductance          ......     357 

Summary  of  Important  Definitions  and  Principles         .          .•„'-.     360 
Problems .          .          .367 

CHAPTER  VIII 
SYMBOLIC  METHOD  OF  TREATING  ALTERNATING  CURRENTS 

203.  Symbolic  Representation  of  a  Vector  .          .          .          .          .371 

204.  Addition  of  Vectors 372 

205.  Substr action  of  Vectors .373 

206.  The  Symbol"  j"  as  a  Multiplier  Signifying  Rotation     .          .          .     374 

207.  Symbolic  Expression  for  a  Vector  Referred  to  Any  Other  Vector  as 

the  Line  of  Reference          .......     375 

208.  Difference  in  Phase  between  two  Vectors  Expressed  in  Symbolic 

Notation  .......  375 

209.  Symbolic  Representation  of  a  Harmonic  Function         .          .          .  376 

210.  Symbolic  Expression  for  the  Derivative  of  a  Harmonic  Function     .  377 

211.  Symbolic   Notation  for  Impedance.     Impedance  as  a  Complex 

Number 378 

212.  Symbolic  Notation  for  Admittance 380 

213.  Division  of  a  Rotating  Vector  by  a  Complex  Number    .         .          .     381 

214.  KirchhofFs  Laws  in  Symbolic  Notation       .         .         .         .         .383 


xvi  CONTENTS 

Art  Page 

215.  Expression  for  Average  Power  in  Symbolic  Notation     .         .         .  385 

216.  Expression  for  Reactive  Power  in  Symbolic  Notation    .         .          .  386 

217.  Expression  for  Power  Factor  in  Symbolic  Notation       .          .          .  387 

218.  Examples  of  Use  of  Symbolic  Method 387 

Summary  of  Important  Definitions  and  Principles         .         .          .391 

Problems 393 

CHAPTER  IX 

THREE-PHASE  ALTERNATING  CURRENTS 

219.  Polyphase  Alternating  Currents          ......  396 

220.  Vector  Sum  of  the  Induced  E.  M.  F.'s  in  the  Three  Windings  of  a 

Three-Phase  Generator  Equals  Zero     .          ...          .          .  397 

221.  Relation  between  Coil  E.  M.  F.  and  E.  M.  F.  between  Terminals 

in  a  Y-Connected  Three-Phase  Generator     .          .          .          .  399 

222.  Currents  from   a  Three-Phase  Generator — Generator  and  Load 

both  Y  Connected .400 

223.  Currents   from    a   Three-Phase   Generator — Generator    A   Con- 

nected and  Load  Y  Connected     ......  402 

224.  Coil  Current  in  a  A-Connected  Generator  for  Balanced  Load          .  404 

225.  Coil  E.  M.  F.  and  Impedance  of  the  Equivalent  Y-Connected  Gen- 

erator and  of  the  Equivalent  Y-Connected  Load    .          .          .  405 

226.  Reduction  of  all  Balanced  Three-Phase  Circuits  to  Equivalent  Y's  406 

227.  Power  in  a  Balanced  Three-Phase  System   .          .         .         ,          .407 

228.  Example  of  Three-Phase  Circuit  Calculation        ....  409 

229.  Rating  of  Three-Phase  Apparatus 410 

230.  Measurement   of  Power  in  a  Three-Phase  Circuit.     Two-Watt- 

meter Method 410 

231.  Two- Wattmeter  Method  Applied  to  a  Balanced  Three-Phase 

System 413 

Summary  of  Important  Definitions  and  Relations          .          .          .  416 

Problems 418 

Appendix  A — Institute  Style  and  Abbreviations  ....  422 

Appendix  B—B.  &  S.  Gauge 425 


Electrical  Engineering 

i 

FUNDAMENTAL  IDEAS  AND  UNITS 

1.  Introduction. — The  fundamental  conceptions  with  which 
we  have  to  start  are  the  ideas  of  space,  time  and  matter.     We 
notice  all  about  us  that  matter  is  changing  its  position  in  space, 
or  moving,  and  that  each  motion  requires  a  certain  interval  of 
time.     By  measuring  the  amount  of  matter  involved,  the  amount 
of  motion  that  takes  place  and  the  time  required,  scientists  have 
found  that  these  quantities   are  invariably  related  in  definite 
quantitative  ways ;  in  other  words,  that  .every  change  in  the 
motion  of  matter  takes  place  in  accordance  with  definite  laws. 

Many  of  the  fundamental  conceptions  used  in  scientific  work 
are  based  on  assumptions  which  are  incapable  of  proof,  but  which, 
on  account  of  their  simplicity  and  plausibility,  we  accept  as  true. 
For  example,  we  accept  as  true  that  the  interval  of  time  required 
for  a  given  change  in  the  position  of  a  given  portion  of  matter  will 
always  be  the  same,  provided  the  conditions  under  which  the 
change  takes  place  are  exactly  duplicated;  again,  we  accept  as 
true  that  whenever  the  velocity  of  a  particle  of  matter  changes, 
this  change  is  due  to  the  influence  of  some  other  particle  or  par^ 
tides  of  matter  or  to  some  agent  associated  with  the  particle 
or  particles.  Such  assumptions  have  been  called  "  Articles  of 
Scientific  Faith." 

In  order  to  express  the  laws  of  nature  in  a  quantitative  man- 
ner, it  is  necessary  to  define  clearly  1.  What  shall  be  taken  as  a 
measure  of  each  quantity,  2.  What  is  meant  by  equal  amounts 
of  this  quantity,  and  3.  What  shall  be  taken  as  the  unit  of  this 
quantity. 

2.  Length. —  Two  straight  lines  which  can  be  superimposed 
one  upon  the  other,  so  that  the  ends  of  the  two  lines  exactly  coin- 
cide, are  said  to  be  equal  in  length.     The  distances  between  any 
two  pairs  of  points  ABsmd  A'B'  respectively  are  said  to  be  equal 
when  the  straight  lines  drawn  between  A  and  B  and  between  A' 
and  Br  are  equal.      If  we  choose  the  distance  between  any  two 
arbitrary  points  as  a  unit  or  standard  of  length,  then  any  other 


2  ELECTRICAL    ENGINEERING 

length  may  be  expressed  as  the  number  of  these  equal  unit  lengths 
into  which  the  line  between  the  two  given  points  may  be  divided. 
The  unit  or  standard  of  length  employed  in  all  scientific  work 
is  the  centimeter,  abbreviated  cm.;  it  is  the  yi^th  portion  of  the 
length  of  a  certain  platinum-iridium  bar  known  as  the  Interna- 
tional Meter  and  preserved  at  the  International  Bureau  of  Weights 
and  Measures  near  Paris;  the  length  of  the  bar  is  measured 
when  it  is  at  the  temperature  of  melting  ice,  i.e.,  at  0°  centigrade. 
Some  of  the  other  common  units  of  length  employed  in  scientific 
and  engineering  work  are  related  to  one  another  as  follows: 
1  meter  =100  centimeters 

1  millimeter  =0.1  centimeter 

1  kilometer  =1000  meters 

1  inch  =2.5400  centimeters 

1  mil  =0.001  inch 

1  foot  =30.480  centimeters 

1  yard  =91.440  centimeters 

1  mile  =5280  feet 

1  mile  =1.6093  kilometers 

1  mile  =1609.3  meters 

3.  Surface.  —  The  unit  of  surface  is  the  area  of  a  square  each 
side  of  which  is  one  unit  in  length;  some  of  the  common  units  of 
surface  are  related  to  one  another  as  follows : 

1  square  inch  =6.4516  square  centimeters 

1  circular  mil  =0.78540  X 10"6  square  inch 

1  circular  mil  =0.00050671  square  millimeter 

1  square  foot  =929.03  square  centimeters 

1  square  yard  =8361.3  square  centimeters 

1  acre  =43,560  square  feet 

1  acre  =4046.9  square  meters 

1  square  mile  =27,878,400  square  feet 

1  square  mile  =640  acres 

1  square  mile  =2.5900  square  kilometers 

4.  Volume.  —  The  unit  of  volume  is  the  volume  of  a  cube  each 
edge  of  which  is  one  unit  in  length.     Some  of  the  common  units 
of  volume  are  related  to  one  another  as  follows : 

1  liter  =1000  cubic  centimeters 

1  cubic  inch  =16.387  cubic  centimeters 

1  cubic  foot  =28,317  cubic  centimeters 

1  cubic  foot  =1728  cubic  inches 


FUNDAMENTAL  IDEAS  AND  UNITS  3 

1  cubic  foot  =7.4805  gallons  (Liquid;  U.  S.) 

1  cubic  yard  =0.76456  cubic  meter 

1  quart  (Liquid;  U.  S.)  =0.94636  liter 

1  gallon  (Liquid;  U.  S.)  =231  cubic  inches 

1  gallon  (Liquid;  U.  S.)  =3.7854  liters 

5.  Angle.  — Let  A B  and  AC  m  Fig.  1  be  two  straight  lines 
intersecting  in  the  point  A.     With  A  as  a  center  and  any  dis- 
tance as  a  radius,  draw  a  circle  about  A  in  the  plane  ABC.     Let 
Bf  and  C'  be  the  points  where  this  circle  cuts  the  lines  A  B  and 
BC  respectively.     Then  the  ratio  of  the  arc  B'C'  to  the  radius 
AB'  or  AC'  is  called  the  angle  between  the  lines  AB  and  BC; 
that  is, 

i      ^  A  r>      arc  B'C'  (1) 

Angle  CAB  = .  W 

AB' 

This  ratio  is  independent  of  the  length  A  B',  since  the  arc  is 
proportional  to  the  radius.  The  unit  angle  as  thus  defined  is 
called  the  "  radian " ;  that  is,  a 
radian  is  the  angle  subtended  by 
an  arc  which  is  equal  to  the  radius. 
Angles  are  also  expressed  in  terms 
of  another  arbitrary  unit  called 
the  "  degree."  One  degree  is  the  A' 
angle  subtended  by  -g^th  part  of 
the  arc  of  a  circle.  Since  the  total  Flg-  1- 

length  of  a  circumference  is  equal  to  2 ?rX( radius),  the  total  plane 
angle  about  a  point  is  equal  to  2?r  radians.  Also,  from  the  defi- 
nition of  the  degree,  the  total  plane  angle  about  a  point  is  360 
degrees.  Hence 

1  radian  =57. 296  degrees. 

The  angle  between  two  planes  which  intersect  in  a  line  M  N 
is  defined  as  the  angle  between  the  two  lines  in  which  these  two 
planes  intersect  a  third  plane  drawn  perpendicular  to  M  N. 

The  angle  made  by  a  given  line  with  any  other  line  which 
it  does  not  intersect  is  defined  as  the  angle  between  the  given 
line  and  a  line  drawn  through  any  point  in  this  line  parallel  to 
the  second  line. 

6.  Time.  — We  accept  as  an  article  of  scientific  faith  that  the 
time  required  for  a  given  change  in  the  position  of  a  given  portion 
of  matter  will  always  be  the  same  provided  all  the  other  conditions 


4  ELECTRICAL    ENGINEERING 

under  which  the  displacement  occurs  are  exactly  duplicated.  The 
simplest  motion  of  this  kind  is  that  of  a  pendulum  suspended  at 
a  given  point  with  reference  to  the  earth  and  oscillating  so  that 
any  given  point  of  the  pendulum  passes  over  the  same  distance 
during  each  oscillation.  We  may  then  take  as  the  numerical 
measure  of  any  interval  of  time  the  number  of  vibrations  made  in 
this  interval  by  such  a  pendulum.  The  unit  or  standard  of  time 
ordinarily  adopted  in  scientific  work  is  the  time  required  for  one 
oscillation  of  a  pendulum,  which,  when  kept  under  absolutely 
constant  conditions,  would  make  86,400  oscillations  in  a  mean 
solar  day;  this  unit  is  called  the  second.  The  solar  day  is  the 
interval  of  time  between  two  successive  transits  of  the  sun  across 
the  meridian  of  the  earth  at  the  point  of  observation;  this  in- 
terval varies  in  length  at  different  times  during  the  year  but 
the  average  length  of  the  interval  for  one  year  is  constant  as  far 
as  we  know.  Some  of  the  common  units  of  time  are  related  to 
one  another  as  follows: 

1  hour  =3600  seconds 

1  day  =86,400  seconds 

1  civil  or  calendar  year  =8760  hours 

1  civil  or  calendar  year          =31,536,000  seconds 

1  solar  year  =365.2422  days 

1  solar  year  =31,556,926  seconds 

1  leap  year  =8784  hours 

7.  Displacement.  —  Let  a  point  which  had  a  position  P  at 
any  instant    have  at    some   later   instant   a    position   Q.     The 
straight  line  drawn  from  P  to  Q  is  called  the  linear  displacement 
of  the  point.     Let  AB  be  any  other  straight  line  in  space  and 
imagine  a  plane  drawn  through  P  and  the  line  AB  and  another 
plane  through  Q  and  the  line  AB;  the  angle  between  the  planes 
P  AB  and  QAB  is  called  the  angular  displacement  of  the  point 
about  the  axis  AB. 

8.  Vectors.  —  The  line  P  Q  representing  the  linear  displace- 
ment of  a  point  has  both  length  and  direction;  the  length  of  the 
line  may  be  represented  by  a  number  and  its  direction  by  the 
angles  made  by   PQ  with  any  three  arbitrarily  chosen  axes  of 
co-ordinates.     In  the  majority  of  problems  that  arise  in  engineer- 
ing work  the  various  points  of  a  body  move  in  parallel  planes ; 
in  this  case  the  direction  of  the  linear  displacement  PQ  of  any 
point  can  be  expressed  numerically  by  the  angle  made  by  the 


FUNDAMENTAL    IDEAS    AND    UNITS  5 

line  PQ  with  an  arbitrary  line  of  reference  fixed  in  any  one  of 
these  planes.  For  example,  in  Fig.  2  let  the  plane  of  the  paper 
be  the  plane  in  which  the  point  moves,  let  OX  be  a  line  fixed 
in  this  plane  and  let  PQ  be  the  linear  displacement  of  the  point. 
Draw  a  line  through  P  parallel  to  OX  and  let  6  be  the  angle 
between  this  line  and  PQ;  then  both  the  amount  and  direc- 


Fig.  2. 

tion  of  the  linear  displacement  of  the  point  are  completely  deter- 
mined when  we  know  the  magnitude  of  the  length  PQ  and  its 
direction.  A  quantity  of  this  kind  which  requires  for  its  com- 
plete representation  a  magnitude  and  a  direction  is  called  a 
vector  quantity  and  the  line  representing  such  a  quantity  is  called 
a  vector.  A  quantity  which  has  magnitude  only,  but  not  direc- 
tion, such  as  time,  mass,  etc.,  is  called  a  scalar  quantity. 

When  a  point  moves  from  P  to  Q  and  then  back  from  Q  to  P, 
the  final  displacement  of  the  point  is  zero;  this  may  be  expressed 
mathematically  by   the  formula  PQ+QP=Q,  or   PQ=-QP. 
That  is,  the  vector  P  Q 
is  equal  but  opposite 
to  the  vector  QP;  the 
vector  Q  P  is  said  to  be 
in  the  opposite   sense 
to  the  vector  PQ.    It 
is  therefore  necessary 
in  dealing  with  vectors 
to    specify   definitely 
the  sense  of  the  vec- 
tor; this  may  be  done       

by  writing  the  letters 
representing  the  ends 
of  the  vector  in  the  order  such 


Fig.  3. 


that    motion  from  the  first  to 


6  ELECTRICAL    ENGINEERING 

the  second  is  in  the  positive  sense  of  the  vector,  or  by  placing 
on  the  line  representing  the  vector  an  arrow  pointing  in  the 
positive  sense  of  the  vector.  In  dealing  with  vectors  which  lie 
in  the  same  or  in  parallel  planes-  (i.e.,  co-planar  vectors),  it  is 
usual  to  select  an  arbitrary  line  drawn  from  an  arbitrary  point 
as  tne  axis  of  reference  and  t«  take  as  the  positive  sense  of  each 
vector  its  sense  away  from  this  arbitrary  point  of  origin.  The 
direction  of  each  vector  may  then  be  expressed  by  the  angle 
measured  around  to  the  vector  in  the  counter-clockwise  direction 
from  the  line  of  reference.  For  example,  in  Fig.  2,  the  vector  PQ 
makes  an  angle  of  30°  with  the  line  of  reference  OX.  In  Fig.  3  the 
vector  P  Q  makes  an  angle  of  330°  with  0  X,  which  is  equivalent 
to  an  angle  of  -  30°  with  0  X. 

9.  Composition  of  Vectors.  —  Since  by  definition  the  linear 
displacement  of  a  point  when  it  moves  from  a  position  P  to  a 
position  Q  is  the  straight  line  PQ,  this  displacement  is  inde- 
pendent of  the  actual  path  over  which  the  point  moves  from 
P  to  Q.  For  example,  in  Fig.  4,  the  point  may  move  in  a  straight 
line  from  P  to  any  other  point  B,  then  in  a  straight  line  to  a  point 
C,  and  finally  to  the  point  Q.  The  lines  PB,  BC  and  CQ  are 

called  the  components  of 
the  vector  PQ,  and  PQ 
is  called  the  resultant  of 
the  vectors  PB,  BC, 
CQ.  PQ  may  therefore 
be  considered  to  be  made 
up  of  any  number  of 
component  vectors,  pro- 
vided that  when  all  of 
these  components  are 
placed  end  to  end  they 
form  a  continuous  path 
from  P  to  Q  such  that 
a  point  moving  from  P  to  Q  over  this  path  moves  in  the  positive 
sense  of  each  component  successively.  Similarly  the  resultant 
of  any  number  of  vectors  PB,  BC,  CQ,  etc.,  is  found  by 
placing  the  lines  representing  these  vectors  end  to  end  in  such 
a  manner  that  a  point  moving  along  the  path  formed  by  these 
lines  always  moves  in  the  positive  sense  of  the  vectors ;  the  line 
drawn  from  the  beginning  of  the  first  vector  to  the  end  of  the 


FUNDAMENTAL  IDEAS  AND  UNITS 


last  of  the  series  is  then  the  resultant.     The  above  facts  may  be 

represented  by  a  formula  thus:    PQ  =  PB+BC+CQ  where  the 

line  over  the  second  term  indicates  that  the  lines  PB,  BC,  and 

CQ  must  be  "  added  " 

in    the    manner    just 

described.       Addition  ^xX"  ^ 

of  this  sort  is  usually 

called     geometric     or 

vector    addition ;    the 

line   over   the   second 

term     then     indicates 

that  PB,  BCandCQ 

must  be  added  geometrically  or  vectorially. 

Similarly,  we  may  subtract  a  vector  PB  from  any  other  vector 

PQ  by  adding  toPQ  a  vector  QB'  equal  to  PB  and  in  the  opposite 

or  negative  sense.     This  may  be  represented  by  a  formula  thus: 

PB'=PQ-PB.     (See  Fig.  5.) 

The  addition  of  two  or  more  co-planar  vectors  may  also  be 

expressed  analytically. 
For  example,  let  it  be  re- 
quired to  find  the  resultant 
R  of  the  three  vectors  A, 
B,  and  C,  Fig.  6.  Choose 
any  arbitrary  line  OX  as 
a  line  of  reference  and  let 
0r,  0a,  0b  and  0C  be  the 
angles  made  by  R,  A,  B, 
and  C  respectively  with 
the  line  OX,  and  call  these 
angles  positive  when  meas- 
ured in  the  counter-clock- 
wise direction  around  from 

OX  and  negative  when  measured  in  the  clockwise  direction.     Then 

the  component  of  R  parallel  to  OX  is 

R  cos  0r  =  A  cos  0a  +  B  cos  0b+C  cos  0C 

and  the  component  of  R  which  makes  an  angle  of  90°  with  OX  is 
R  sin  0r=A  sin  0a+ B  sin  0b+C  sin  0C 

Hence  the  length  of  R  is 


Fig.  6. 


8 


ELECTRICAL    ENGINEERING 


and  the  angle  which  R  makes  with  OX  is  Or)  where 
A  sin  ft  +  5  sin  0b+C  sin  9C 


tan  Or  = 

For  example,  let 

A  =3 

B=2 

C=5 
Then 


A  cos  0n+B  cos  ft  +  C  cos  6f 


0a=30° 


ft.  =60 


(26) 


=(3  X  0.8664-  2  X  0.707+  5  X  0.5)2+(3  X 0.5+  2  X  0.707+5  X 0.866)2=  9.74 


tan  0r 

0r 


7.244 
:  6.512 

=48.05° 


=  1.1124 


We  shall  see  later  on  that  many  of  the  quantities  met  with  in 
engineering  problems,  such  as  velocity,  force,  moments,  electric 
current,  etc.,  can  be  represented  by  vectors.  In  certain  cases 
it  will  be  seen  that  it  will  be  unnecessary  to  specify  the  location 
of  the  vector,  but  that  any  two  vectors  which  are  equal  in  length 
and  are  parallel  may  be  considered  equivalent;  in  other  cases 
we  shall  find  that  we  may  consider  two  equal  and  parallel  vectors 
as  equivalent  only  when  they  lie  in  the  same  plane,  or  in  the 
same  line,  or  it  may  be  that  the  vector  cannot  be  considered 
equivalent  to  any  other  vector.  When  the  location  of  a  vector  is 
thus  limited  it  is  said  to  be  localized  in  a  plane,  on  a  line  or  at  a 
point,  as  the  case  may  be.  The  above  laws  for  the  composition 

and  resolution  of  vectors  apply 
to  localized  vectors  only  in  case 
the  vectors,  or  vectors  equiv- 
alent to  them,  meet  in  a  point. 
The  angular  displacement 
of  a  point  about  any  axis  is  a 
quantity  which  requires  for  its 
representation  a  vector  local- 
ized in  a  line.  For  example,  let 
P  and  Q,  lying  in  the  plane  of 
Fig  7>  the  paper,  be  the  initial  and 

final  positions  of  the  point;  let 

the  axis  of  rotation  be  a  line  drawn  perpendicular  to  this  plane  at 
A.     By  definition,  the  angular  displacement  of  the  point  about  this 


FUNDAMENTAL   IDEAS  AND   UNITS  9 

axis  is  then  the  angle  PAQ.  A  line  drawn  in  the  axis  through  A  hav- 
ing a  length  equal  numerically  to  the  angle  PAQ  will  serve  to  repre- 
sent both  the  axis  of  rotation  and  the  numerical  value  of  the  angular 
displacement.  To  represent  the  sense  of  the  rotation,  i.e.,  whether 
from  P  to  Q  or  from  Q  to  P,  it  is  customary  to  choose  arbitrarily  the 
positive  sense  of  the  axis,  and  then  call  the  rotation  positive  when 
the  motion  of  the  point  is  in  a  clockwise  or  right-handed  direction 
when  viewed  by  a  person  looking  in  the  sense  of  this  line.  In 
the  case  illustrated,  if  we  choose  the  positive  sense  of  the  axis 
to  be  toward  the  reader  the  line  representing  the  angular  dis- 
placement from  P  to  Q  will  be  drawn  upward  perpendicular  to  the 
plane  of  the  paper;  the  line  representing  the  displacement  from  Q 
to  P  will  be  drawn  downward.  The  angular  displacement  of  a 
point  about  any  axis  is  therefore  completely  defined  by  a  line  having 
(1)  a  definite  length,  (2)  a  definite  direction,  and  (3)  localized  in  the 
axis  of  rotation.  We  may  also  represent  the  displacement  of  the 
point  from  the  position  P  to  the  position  Q  by  an  angular  displace- 
ment about  any  other  axis,  such  as  an  axis  through  B  parallel  to  the 
axis  through  A,  but  the  vector  representing  this  angular  displace- 
ment about  the  axis  through  B  will  not  be  equal  to  the  vector  repre- 
senting the  angular  displacement  about  the  axis  through  A,  since 
the  angles  PAQ  and  PBQ  are  not  equal.  Again,  two  equal  and 
parallel  vectors  do  not  represent  equivalent  angular  displace- 
ments unless  these  two  vectors  lie  in  the  same  line. 

10.  Velocity.  —  Let,  in  any  small  interval  of  time  dt,  a  point 
P  be  displaced  a  distance  dl  with  respect  to  any  other  point  0; 

then  the  limiting  value  at  any  instant  of  the  ratio  —  ,  when  dt  is 

dt 

taken  extremely  small,  is  called  the  linear  velocity  of  the  point  P 
at  that  instant  relative  to  the  point  0.  Representing  linear 
velocity  by  v,  we  have 

dl  , 


When  the  point  moves  in  a  straight  line  over  equal  distances  in 
equal  small  intervals  of  time  its  linear  velocity  is  said  to  be  uni- 
form; in  this  case  the  linear  velocity  of  the  point  may  be  defined 
as  the  linear  displacement  of  the  point  in  unit  time.  Since  linear 
displacement  is  a  vector  quantity,  i.e.,  has  both  magnitude  and 
direction,  and  time  is  a  scalar  quantity,  linear  velocity  is  also  a 


10  ELECTRICAL   ENGINEERING 

vector  quantity;  for  a  vector  quantity  divided  by  a  scalar  is  a 
vector  quantity.  Therefore  a  change  either  in  the  magnitude  or 
in  the  direction  of  the  linear  velocity  constitutes  a  change  in  the 
linear  velocity.  The  magnitude  of  the  linear  velocity  of  a  point  is 
frequently  called  the  linear  speed  of  the  point;  that  is,  the  linear 
speed  of  a  point  is  simply  a  number  expressing  the  distance  over 
which  the  point  moves  in  unit  time;  linear  speed  is  therefore  a 
scalar  quantity.  For  example,  a  point  moving  in  a  circle  in  such 
a  manner  that  in  equal  small  intervals  of  time  it  passes  over  equal 
distances  measured  along  the  circumference  of  the  circle,  is  said  to 
be  moving  with  a  constant  speed;  its  velocity,  however,  is  changing 
at  every  instant,  since  the  direction  of  motion  is  continually 
changing. 

Linear  speeds  may  be  expressed  in  various  units,  such  as  the 
number  of  centimeters  per  second,  feet  per  second,  miles  per  hour, 
etc.  The  more  common  units  are  related  as  follows : 

1  kilometer  per  hour  =0.91 134  foot  per  second 

1  kilometer  per  hour  =54.681  feet  per  minute 

1  mile  per  hour  =0.44704  meter  per  second 

1  mile  per  hour  =1.46667  feet  per  second 

1  mile  per  hour  =88  feet  per  minute 

Angular  velocity  is  defined  in  exactly  the  same  way  as  linear 
velocity,  i.e.,  the  angular  velocity  of  a  point  about  any  axis 

18  de 

O)= 

dt  (4) 

where  d6  is  the  angular  displacement  of  the  point  about  that 
axis  in  the  small  interval  of  time  dt.  Angular  velocity  is  rep- 
resented by  a  localized  vector  in  the  same  way  that  angular 
displacement  is  represented  by  a  localized  vector.  The  term 
angular  speed  is  used  to  express  the  magnitude  of  the  angular 
velocity  in  the  same  way  that  linear  speed  is  used  to  express 
the  magnitude  of  linear  velocity.  Angular  speeds  may  be  ex- 
pressed in  degrees  per  unit  time,  radians  per  unit  time  or  revo- 
lutions per  unit  time.  The  more  common  units  are  related  as 
follows : 

1  radian  per  second        =57.296  degrees  per  second 
1  radian  per  second        =0.159155  revolution  per  second 
11.   Acceleration.  — The  rate  of  increase  of  linear  velocity  with 
respect  to  time  is  called  the  linear  acceleration;  i.e.,  when  the  linear 


FUNDAMENTAL  IDEAS  AND   UNITS  11 

velocity  of  a  point  increases  by  a  small  amount  dv  in  a  small  inter- 
val of  time  dt,  then  the  linear  acceleration  is 

_dv 

=~d~t'  (5) 

Note  that  dv  is  the  difference  between  the  two  vectors  represent- 
ing the  linear  velocities  at  the  beginning  and  end  of  the  small  in- 
terval of  time  dt,  and  that  in  general  the  direction  of  the  vector 
representing  this  difference  will  have  no  fixed  relation  to  the 
direction  of  the  vectors  representing  these  two  velocities.  How- 
ever, when  the  direction  of  the  line  of  motion  does  not  change, 
i.e.,  when  these  two  velocities  are  in  the  same  direction,  the  accel- 
eration will  likewise  be  in  the  same  or  the  opposite  direction  to 
the  velocity.  In  this  case  the  linear  acceleration  is  equal  to  the 
second  derivative  of  the  displacement,  that  is, 

JPl 

~dP  (6) 

where  dl  is  the  linear  displacement  in  time  dt.  On  the  other 
hand,  when  the  speed  remains  constant  (i.e.,  only  the  direction 
of  the  velocity  changes)  it  can  be  readily  shown  that  this  vector 
difference,  and  therefore  the  direction  of  the  acceleration  a,  is  at 
every  instant  perpendicular  to  the  path  of  motion  and  is  toward 
the  center  of  curvature  of  this  path,  and  is  equal  to  the  square 
of  the  linear  speed  s  divided  by  the  radius  of  curvature  r  of  this 
path;  that  is, 

s2 

.       a=V  (7) 

The  commonest  linear  acceleration  with  which  we  have  to  deal 
is  the  acceleration  of  falling  bodies,  or,  as  it  is  commonly  called,  the 
acceleration  "  due  to  gravity."  This  acceleration  is  constant  for 
all  kinds,  shapes  and  sizes  of  bodies  falling  in  a  vacuum,  for  any 
given  place  on  the  earth's  surface,  but  varies  slightly  with  the 
latitude  and  with  the  altitude  of  the  point  of  observation.  At 
mean  sea  level  and  45°  latitude  its  value,  as  determined  by 
Helmert  in  1884,  is  980.5966  centimeters  per  second.  This  value 
is  sometimes  used  as  the  unit  of  acceleration;  it  is  called  the  ac- 
celeration of  gravity  and  is  represented  by  the  symbol  g.  Other 
common  units  of  linear  acceleration  are  related  as  follows : 


12  ELECTRICAL  ENGINEERING 

1  kilometer  per  hour  per  second 

=27.778  centimeters  per  second  per  second 
=0.62137  mile  per  hour  per  second 
=0.028327  gravity 
1  mile  per  hour  per  second 

=44.704  centimeters  per  second  per  second 
=  1.46667  feet  per  second  per  second 
=0.045589  gravity 
=3600  miles  per  hour  per  hour 

Gravity  =980.5966  centimeters  per  second  per  second 
Gravity  =32.172  feet  per  second  per  second 
The  variation  of  gravity  with  altitude  and  location  is  very 
slight,  and  the  approximate  values  981  centimeters  per  second  per 
second  and  32.2  feet  per  second  per  second  are  as  a  rule  sufficiently 
accurate  for  engineering  work,  independent  of  altitude  or  location. 
The  angular  acceleration  of  a  point  is  similarly  defined  as  the 
rate  of  change  of  its  angular  velocity.     In  case  the  point  is  rotating 
in  a  circle  about  a  fixed  axis,  its  angular  acceleration  may  be 
defined  as  the  rate  of  change  of  its  angular  speed  about  this  axis. 
If  CD  is  the  angular  speed  then  the  angular  acceleration  is 

_do) 
=~di'  (8) 

T    /} 

Since  a>  = — ,  where  dO  is  the  angular  displacement  in  time  dt, 
dt 

we  also  have,  under  the  same  conditions,  that 

_<P6 

~df'  (9) 

12.  Mass.  —  The  quantity  of  matter  in  a  body  or  its  mass 
can  be  defined  only  in  terms  of  some  effect  produced  on  the  body  by 
some  other  body  or  bodies  exterior  to  it.  It  has  been  found  by 
experiment  that  two  bodies,  which  appear  to  our  senses  to  be 
identical  in  every  respect,  will  exactly  counter-balance  each  other 
when  suspended  one  from  each  end  of  an  equal-armed  balance  in  a 
vacuum.  We  may,  then,  go  a  step  further  and  define  the  mass  of 
any  two  bodies  as  equal  irrespective  of  their  volume,  shape  or 
chemical  composition,  if,  when  they  are  suspended  simultaneously 
in  a  vacuum,  one  from  each  end  of  an  equal-armed  balance,  there 
is  no  tipping  of  the  beam  of  the  balance  from  its  original  position. 
This  criterion  for  the  equality  of  two  masses  holds  only  in  case  the 


FUNDAMENTAL  IDEAS  AND   UNITS  13 

bodies  and  the  balance .  are  neither  electrically  charged  nor  mag- 
netised, and  both  bodies  are  supported  at  the  same  distance 
from  the  earth,  and  the  equilibrium  of  the  balance  is  not  affected 
by  the  presence  of  any  other  bodies  (except  the  earth)  in  the 
vicinity. 

This  is  an  entirely  arbitrary  definition,  but  it  has  been  found 
that  mass  as  thus  defined  is  a  fundamental  property  of  matter. 
Any  arbitrary  portion  of  matter  may  be  taken  as  the  unit  of  mass; 
the  mass  of  any  given  portion  of  matter  may  then  be  expressed  as 
the  number  of  such  equal  units,  which  taken  together,  and  sus- 
pended from  one  arm  of  an  equal-armed  balance  in  a  vacuum,  will 
just  counter-balance  the  given  body  suspended  from  the  other  arm. 
Note  that  mass  is  a  scalar  quantity. 

The  unit  or  standard  of  mass  adopted  in  all  scientific  work  is 
the  gram,  abbreviated  g;  it  is  the  TcWth  portion  of  a  certain 
platinum-iridium  cylinder,  known  as  the  International  Kilogram 
and  preserved  at  the  International  Bureau  of  Weights  and  Meas- 
ures near  Paris.  Some  of  the  common  units  of  mass  are  related 
to  one  another  as  follows: 

1  metric  ton  =1000  kilograms 

1  centigram  =0.01  gram 

1  milligram  =0.001  gram 

1  pound  (avoirdupois)  =453.592  grams 

1  short  ton  =2000  Ibs. 

1  short  ton  =907.185  kilograms 

1  short  ton  =0.907185  metric  ton 

1  long  or  gross  ton  =2240  Ibs. 

13.  C.  G.  S.  or  Absolute  System  of  Units.  —  We  have  seen  that 
the  standard  units  adopted  in  scientific  work  for  the  measurement 
of  the  fundamental  quantities  of  length,  mass  and  time  are  the 
centimeter,  gram  and  second.     Units  for  the  measurement  of  all 
other  quantities  such  as  surface,  volume,  velocity,  acceleration, 
force,  etc.,  can  be  expressed  in  terms  of  these  units;  such  units  are 
called  derived  units  in  contradistinction  to  the  three  fundamental 
or  absolute  units  of  length,  mass  and  time.     The  system  of  units 
derived  from  the  units  of  centimeter,  gram  and  second  is  known  as 
the  absolute  system,  or  the  c.  g.  s.  system,  from  the  initials  of  the 
three  fundamental  units. 

14.  Density  and  Specific  Gravity.  —  The  density  of  a  uniform 
substance  is  defined  as  the  mass  of  the  substance  per  unit  volume. 


14  ELECTRICAL  ENGINEERING 

In  the  c.  g.  s.  system  density  is  the  weight  in  grams  of  one  cubic 
centimeter  of  the  substance.  When  the  substance  is  not  uniform, 
its  density  at  any  point  is  denned  as  the  mass  of  an  infinitesi- 
mally  small  volume  taken  about  the  point  divided  by  this  volume ; 
i.e.,  calling  dv  the  volume  and  dm  the  mass  of  this  volume,  the 
density  is 

dm 

"dv  '  (10) 

The  specific  gravity  of  a  substance  is  defined  as  the  ratio  of  the 
weight  of  a  given  volume  of  the  substance  to  an  equal  volume  of 
water  at  standard  temperature.  Sixty-two  degrees  Fahrenheit  is 
usually  taken  as  the  standard  temperature,  although  there  is  no 
general  agreement  on  this  point.  Density  on  the  c.  g.  s.  system 
and  specific  gravity  are  practically  numerically  equal. 

16.  Center -of  Mass.  — A  body  of  mass  M  which  has  any  size  or 
shape  may  be  considered  as  made  up  of  a  number  of  small  par- 
ticles of  masses  mlj  m2,  ms,  etc.,  such  that  mlJrm2-\-m5-\ =M. 

These  particles  may  be  considered  as  small  as  we  wish,  that  is 
we  may  consider  each  particle  so  small  that  it  occupies  but  a  point 
in  space.  If  we  consider  three  mutually  perpendicular  planes 
X,  Y,  Z,  fixed  in  space,  and  represent  by  xlt  yl}  and  zl  the 
perpendicular  distances  of  the  particle  ml  from  these  planes 
respectively,  and  by  xa,  y2,  and  z2  the  perpendicular  distances  of 
the  particle  m2  from  these  three  planes  respectively,  and  so  on 
for  the  other  particles,  then  the  point  whose  distances  from  these 
three  planes  are  respectively 

w     +  mx 


M 

-  (11) 


M 

2  —    !_ 


M 

is  defined  as  the  center  of  mass  of  the  body.  The  center  of  mass 
of  a  body  is  therefore  the  point  the  distance  of  which  from  each 
of  three  mutually  perpendicular  planes  is  the  average  distance  of 
the  matter  in  the  body  from  each  of  these  planes.  It  can  be  shown 
that  the  position  of  the  center  of  mass  of  a  body  relative  to  any 


FUNDAMENTAL   IDEAS   AND   UNITS  15 

point  in  the  body  is  independent  of  the  position  of  the  planes  of 
reference. 

The  center  of  mass  of  a  system  of  any  number  of  bodies  is 
defined  in  exactly  the  same  manner,  except  that  M  in  this  case 
is  taken  as  the  total  mass  of  all  the  bodies. 

1 6.  Linear  Momentum  and  Moment  of  Momentum.  —  When 
a  body  as  a  whole  is  in  motion,  or  when  there  is  any  relative 
motion  of  parts  of  the  body,  the  various  points  of  the  body  will 
in  general  move  with  different  velocities.  It  is  therefore  con- 
venient, in  analysing  the  motion  of  a  system  of  bodies,  to  con- 
sider each  body  as  made  up  of  a  number  of  individual  particles 
and  to  take  these  particles  so  small  that  the  mass  of  each  may 
be  considered  as  occupying  but  a  point  in  space.  The  product  of 
the  mass  (m)  of  each  particle  times  its  linear  velocity  (v)  is  defined 
as  the  linear  momentum  of  the  particle,  i.e.,  linear  momentum  =mv. 
Linear  momentum  is  a  vector  quantity,  since  v  is  a  vector  quan- 
tity and  m  is  a  scalar  quantity.  The  vector  sum  of  the  linear 
momenta  of  all  the  particles  of  a  rigid  body  is  called  the  total 
linear  momentum  of  the  body.  From  the  above  definition  of 
center  of  mass  it  can  be  shown  that  the  total  linear  momentum 
of  a  body  is  equal  to  its  total  mass  times  the  linear  velocity  of 
its  center  of  mass. 

Consider  a  fixed  axis,  and  a  particle  of  mass  m  at  a  distance 
r  from  the  axis,  and  let  the  particle  be  moving  with  a  velocity  v. 
Then  the  product  of  m,  r  and  that  component  u  of  the  velocity 
v  perpendicular  to  the  plane  passing  through  the  particle  and 
the  axis,  is  defined  as  the  moment  of  momentum  of  the  particle 
about  the  fixed  axis;  i.e.,  moment  of  momentum  —mm.  The 
moment  of  momentum  of  a  particle  is  to  be  taken  positive  if 
the  particle  moves  in  a  clockwise  direction  as  seen  by  an  observer 
looking  along  the  axis  in  the  positive  sense,  negative  if  in  the 
opposite  direction.  The  component  u  of  the  linear  velocity  of 
a  particle  at  right  angles  to  a  plane  passing  through  the  particle 
and  any  fixed  axis,  is  equal  to  the  product  of  the  distance  r  of 
the  particle  from  the  axis  times  the  angular  velocity  a>  of  the 
particle  about  this  axis;  therefore  the  moment  of  momentum 
of  the  particle  may  also  be  written  mr^a).  In  the  case  of  a  rigid 
body  each  particle  of  which  has  the  same  angular  velocity  o)  about 
a  given  axis,  the  moment  of  momentum  about  this  axis  is 
equal  to  wSmr3,  the  summation  including  all  the  particles  of  the 


16  ELECTRICAL  ENGINEERING 

body.  The  quantity  ^  mr2  is  called  the  moment  of  inertia  of  the 
body  about  the  given  axis  of  rotation,  and  is  usually  written  7; 
then  7=Swr2  (12) 

The  moment  of  momentum  of  a  solid  body  rotating  in  this  man- 
ner is  called  the  angular  momentum  of  the  body ;  angular  momen- 
tum then  equals  7w.  The  moment  of  inertia  of  a  body  of  given 
dimensions  and  given  distribution  of  material  about  a  given 
axis  may  be  written  Mk*  where  M  is  the  total  mass  of  the  body 
and  k  is  a  length  such  that  ^  , 

F=*^.  (I2a) 

M 

The  length  k  is  called  the  radius  of  gyration  of  the  body  about  the 
given  axis. 

17.  Conservation  of  Mass,  Conservation  of  Linear  Momentum, 
Conservation  of  Moment  of  Momentum.  —  It  has  been  found  that 
every  phenomenon  of  nature,  which  has  so  far  been  tested  by 
experiment,  takes  place  in  such  a  way  that  the  three  following 
conditions  are  invariably  satisfied : 

1.  Matter  cannot  be  created  or  destroyed.     As  a  consequence 
of  this  condition,  if  any  number  of  bodies  are  kept  entirely  sepa- 
rate from  the  rest  of  the  universe,  for  example,  in  a  closed  vessel 
through  the  walls  of  which  no  matter  can  pass,  then  the  total  mass 
of  these  bodies  must  likewise  remain  constant,  irrespective  of  any 
changes  that  may  take  place  in  these  bodies.     This  condition  is 
known  as  the  law  or  principle  of  the  conservation  of  mass. 

2.  When  the  linear  momentum  of  one  or  more  bodies  changes 
relative  to  any  fixed  point,  then  there  must  be  an  equal  and 
opposite  change  in  the  linear  momentum  of  some  other  body  or 
bodies  relative  to  this  point.     This  condition  is  known  as  the 
law  or  principle  of  the  conservation  of  linear  momentum. 

3.  When  the  moment  of  momentum  of  any  body  or  bodies 
about  any  fixed  axis  changes,  then  there  must  be  an- equal  and 
opposite  change  in  the  moment  of  momentum  of  some  other  body 
or  bodies  about  this  same  axis.     This  condition  is  known  as  the 
law  or  principle  of  the  conservation  of  moment  of  momentum. 

18.  Force.  —  In  general  terms,  a  force  is  that  which  produces 
or  tends  to  produce  a  change  in  the  motion  of  a  body.     Since, 
by  the  principle  of  the  conservation  of  linear  momentum,  any 
change  in  the  motion  of  a  particle  A  is  accompanied  by  a  change 
in  the  motion  of  some  other  particle  or  particles  B,  it  is  convenient 


FUNDAMENTAL   IDEAS  AND   UNITS  17 

to  consider  the  force  acting  on  A  as  due  to  the  presence  of  the 
particle  or  particles  B.  Or,  we  may  say  that  the  change  in  the 
motion  of  A  is  due  to  a  force  produced  on  A  by  the  particle  or 
particles  B.  In  other  words,  we  may  consider  the  changes  in 
motion  of  material  particles  as  being  due  to  a  property  possessed 
by  these  particles  themselves.*  We  take  arbitrarily  as  the 
measure  of  the  force  with  which  a  particle  A  is  acted  upon  by 
a  particle  B  the  time  rate  of  change  of  the  linear  momentum  of 
A  relative  to  any  fixed  point,  when  the  relative  motion  of  the  two 
particles  with  respect  to  each  other  is  unaffected  by  the  presence 
of  any  other  particle  or  particles ;  the  direction  of  the  force  on 
A  due  to  the  particle  B  is  defined  as  the  direction  of  the  rate  of 
change  of  the  linear  momentum  of  A.  From  the  principle  of 
the  conservation  of  linear  momentum  we  then  conclude  that  the 
force  with  which  B  is  acted  upon  by  A  is  equal  and  opposite 
to  the  force  with  which  A  is  acted  upon  by  B,  that  is,  "  action 
and  reaction  are  equal  and  opposite."  When  the  masses  of 
the  two  particles  A  and  B  remain  unchanged,  the  time  rate  of 
change  of  linear  momentum  of  each  particle  is  equal  to  the  product 
of  its  mass  and  its  acceleration,  i.e.,  the  force  with  which  A  acts 
on  B  is  ma  and  the  force  with  which  B  acts  on  A  is  m^,  where 
m  and  m^  are  the  masses  of  A  and  B  respectively  and  a  and  at  are 
the  accelerations  of  A  and  B  respectively.  The  direction  of  the 
force  acting  on  A  is  then  the  direction  of  the  linear  acceleration 
of  A,  and  the  direction  of  the  force  acting  on  B  is  the  direction 
of  the  linear  acceleration  of  B.  The  acceleration  of  A  will  then 

*This  conception  of  the  something  that  causes  changes  in  the  motion  of 
jnatter  as  localized  in  material  particles  merely  gives  us  a  convenient  way 
of  describing  experimentally  observed  facts.  It  is  also  conceivable  that  the 
changes  in  the  motion  of  matter  are  due,  not  to  a  propert}r  inherent  in  mat- 
ter itself,  but  to  a  property  possessed  by  the  medium  or  the  "  ether' '  in 
which  the  particles  of  matter  are  immersed.  However,  a  particle  of  matter 
is  a  tangible  thing,  whereas  the  existence  of  the  ether  is  purely  hypothetical. 
For  this  reason  we  shall  adhere  to  the  older  conception  that  every  force, 
whether  it  be  gravitational,  electric,  or  magnetic,  has  its  origin  in  a  material 
particle  and  that  it  can  make  its  influenca  felt  on  other  particles  at  a  distance 
from  it.  As  far  as  engineering  is  concerned,  we  need  not  concern  ourselves 
with  the  mechanism  by  which  this  action  takes  place,  about  which,  at  best, 
we  can  only  theorize.  Analogies,  where  they  help  one  to  form  a  mental 
picture  of  how  certain  effects  might  be  produced,  are  extremely  useful  and 
will  be  frequently  employed;  but  it  must  be  remembered  that  analogies  do 
not  explain  anything. 


18  ELECTRICAL  ENGINEERING 

be  in  the  opposite  sense  to  the  acceleration  of  B,  and  the  ratio 
of  the  acceleration  of  A  to  that  of  B  will  be  equal  to  the  inverse 
ratio  of  the  respective  masses.  Since  acceleration  is  a  vector 
quantity,  force  is  likewise  a  vector  quantity;  therefore,  when 
there  are  several  forces  acting  on  a  particle,  the  resultant  force 
on  the  particle  is  the  vector  sum  of  all  the  forces  acting;  this 
resultant  force  is  also  equal  to  the  product  of  the  mass  of  the 
particle  by  its  acceleration  in  the  direction  of  the  force.  The 
line  through  a  particle  in  the  direction  of  the  force  acting  on  it 
is  called  the  line  of  action  of  the  force. 

The  unit  of  force  in  the  c.  g.  s.  system  of  units  is  that  force 
which  will  give  unit  linear  acceleration  (one  centimeter  per  second 
per  second)  to  a  mass  of  one  gram;  this  unit  force  is  called  the  dyne. 
A  similar  unit  in  the  foot-pound-second  system,  called  the  poundal, 
is  sometimes  used;  it  is  defined  as  the  force  which  will  accelerate 
one  pound  one  foot  per  second  per  second. 

It  is  a  matter  of  experience  that  each  particle  of  matter  which 
has  neither  electric  nor  magnetic  properties  (to  be  described  later), 
when  allowed  to  fall  in  a  vacuum  from  any  given  height  at  any 
point  near  the  earth's  surface,  falls  to  the  earth  in  a  vertical  line 
with  a  constant  acceleration  independent  of  the  size,  shape,  or 
material  of  the  body.  We  therefore  say  that  the  earth  exerts 
a  force  on  each  particle  of  the  body  proportional  to  its  mass, 
and  that  therefore  the  total  force  with  which  the  earth  "  attracts  " 
a  given  body  is  equal  to  the  total  mass  of  the  body  multiplied  by 
its  acceleration  when  falling  freely  to  the  earth  in  a  vacuum. 
We  have  seen  (Article  11)  that  the  acceleration  "  due  to  gravity  " 
is  constant  for  any  given  point  with  reference  to  the  earth's 
surface,  but  varies  slightly  with  the  location  and  also  with 
the  elevation  above  the  earth.  For  most  practical  purposes  the 
acceleration  due  to  gravity  may  be  taken  as  981.  The  attraction 
of  the  earth  on  a  body  offers  a  ready  means  for  measuring  a  force, 
since  it  is  only  necessary  to  balance  the  force  to  be  measured 
against  the  force  exerted  by  the  earth  on  a  known  mass,  or  by 
measuring  the  change  produced  by  the  force  to  be  measured  in 
the  shape  of  some  body,  e.g.,  a  spiral  spring,  which  has  been  pre- 
viously "calibrated  "  by  suspending  known  masses  from  it.  These 
are  the  usual  ways  of  measuring  forces,  since  the  determination 
of  the  acceleration  due  to  any  other  force  than  "  gravity  "  is 
extremely  difficult. 


FUNDAMENTAL   IDEAS  AND   UNITS  19 

When  forces  are  measured  in  this  way,  they  are  usually  ex- 
pressed in  terms  of  units  which  are  given  the  same  name  as  the 
corresponding  units  of  mass.  For  example,  by  a  force  of  one 
gram  is  meant  the  force  with  which  the  earth  attracts  one  gram 
at  sea  level  and  45°  latitude;  this  is  equal  to  a  force  of  980.5966 
dynes,  or  approximately  981  dynes.  Similarly,  a  force  may  be 
expressed  as  so  many  kilograms  or  so  many  pounds.  It  should  be 
noted  that  forces  specified  in  this  way  are  not  absolutely  definite 
unless  the  place  at  which  the  force  is  measured  is  also  stated;  the 
variation  at  ordinary  places  of  observation,  however,  is  so  slight 
that  it  is  negligible  in  ordinary  engineering  work. 

The  relations  between  grams,  kilograms,  pounds,  tons,  etc.,  are 
the  same  whether  these  quantities  are  considered  as  masses  or 
forces;  these  relations  are  given  in  Article  12.  The  units  kilograms 
and  pounds  as  forces  are  related  to  the  dyne  and  the  poundal 
at  sea  level  and  45°  latitude  as  follows : 

1  kilogram  =980,596.6  dynes 

1  kilogram  =70.927  poundals 

1  pound  =444,791  dynes 

1  pound  =32.172  poundals 

19.  Moment  of  Force  or  Torque.  —  Consider  a  fixed  axis  and 
a  particle  at  a  distance  r  from  this  axis.  Let  /„  be  the  com- 
ponent of  the  force  /  acting  on  this  particle  perpendicular  to  the 
plane  passing  through  the  particle  and  the  axis.  The  product 
f0r  is  called  the  moment  of  the  force  f  about  this  axis.  It  can  be 
deduced  from  the  principle  of  the  conservation  of  moment  of 
momentum  that,  when  there  are  any  number  of  external  forces 
acting  on  a  system  of  particles,  the  rate  of  change  of  the  total 
moment  of  momentum  of  the  system  about  any  fixed  axis  is 
equal  to  the  algebraic  sum  of  the  moments  of  all  the  forces  about 
this  axis,  irrespective  of  any  forces  which  the  individual  particles 
may  exert  on  one  another. 

When  a  rigid  body  moves  in  such  a  manner  that  each  point 
has  at  any  instant  the  same  angular  velocity  about  a  given  axis, 
the  rate  of  change  of  the  angular  momentum  of  the  body  about 
this  axis  is  equal  to  the  product  of  the  moment  of  inertia  (/)  of  the 
body  about  the  given  axis  and  the  angular  acceleration  (a)  about 
this  axis;  hence, 

/a  =  2/0r  (13a) 


20  ELECTRICAL  ENGINEERING 

The  moment  of  a  force  about  any  axis  is  frequently  called  the 
torque  about  this  axis;  we  then  have  that  the  algebraic  sum  of  all 
the  torques  (Tlf  T.2,  T3,  etc.)  about  any  axis  is  equal  to  the  product 
of  the  moment  of  inertia  (7)  of  the  body  about  this  axis  and  the 
angular  acceleration  (a)  about  this  axis,  that  is 

Ia  =  T1+  T2  +  Ts  +  —.  (136) 

When  the  line  of  action  of  any  force  passes  through  the  axis,  the 
torque  corresponding  to  this  force  is  of  course  zero.  When  two 
forces  are  acting  on  a  body,  the  condition  for  constant  angular 
velocity  is  that  the  corresponding  torque  be  equal  and  opposite. 

The  unit  torque  in  the  c.  g.  s.  system  of  units  is  the  torque  pro- 
duced on  a  particle  by  a  force  of  one  dyne  acting  perpendicular  to 
the  plane  determined  by  the  position  of  the  particle  and  the 
axis  of  rotation  and  at  a  distance  of  one  centimeter  from  the 
latter  —  this  unit  is  called  the  centimeter-dyne.  Other  common 
units  of  torque  are  the  pound-foot  and  the  centimeter-gram.  The 
relations  of  these  units  one  to  another  are  the  same  as  for  the 
units  of  energy  (see  Article  21)  having  corresponding  names. 

20.  Motion  of  a  System  of  Particles  Acted  upon  by  Several 
Forces. — When  only  one  of  a  system  of  particles  is  acted  upon  by  a 
force  external  to  the  system  (for  example,  a  stretched  string  at- 
tached to  a  point  in  a  solid  body)  the  particle  on  which  the  force 
is  acting  will  in  general  exert  a  force  on  the  other  particles  of  the 
body  and  those  particles  in  turn  will  each  exert  a  force  on  the  par- 
ticle to  which  the  force  is  applied.  It  can  readily  be  shown  that, 
as  a  consequence  of  the  principle  of  the  conservation  of  linear 
momentum,  the  resultant  acceleration  of  the  system  of  particles  in 
this  case  will  be  such  that  the  center  of  mass  of  the  system  will  be 
given  an  acceleration  equal  to  the  external  force  acting  on  the 
particle  divided  by  the  total  mass  of  the  body;  that  is,  a  single 
external  force  acting  on  any  particle  of  a  system  will  produce  the 
same  acceleration  of  the  center  of  mass  of  the  system  as  would  be 
produced  by  the  same  force  acting  on  a  single  particle  located  at 
the  center  of  mass  of  system  and  having  a  mass  equal  to  the 
total  mass  of  the  system.  In  the  case  of  several  external  forces 
acting  on  the  system,  the  acceleration  of  the  center  of  mass  of 
the  system  will  be  equal  to  the  resultant  force  divided  by  the 
total  mass,  or  calling  M  the  mass  of  the  system,  F  the  resultant 
of  all  the  external  forces  acting  on  it,  and  A  the  linear  acceleration 
of  the  center  of  mass,  then 


FUNDAMENTAL   IDEAS   AND   UNITS  21 

F  =  MA.  (14) 

In  general,  each  particle  of  the  system  will  also  have  its  angular 
velocity  about  any  axis  changed  due  to  the  action  of  a  force  on  any 
one  particle.  In  the  case  of  a  rigid  body,  it  can  be  shown  that  in 
addition  to  the  change  in  the  velocity  of  the  center  of  mass  pro- 
duced by  this  force,  each  particle  of  the  body  wih1  be  given  an 
angular  acceleration  about  an  axis  through  the  center  of  mass  per- 
pendicular to  the  plane  determined  by  the  center  of  mass  and  the 
line  of  application  of  the  force,  and  that  this  angular  acceleration 
will  be  equal  to  the  moment  of  the  force  about  this  axis  divided 
by  the  moment  of  inertia  of  the  body  about  this  axis.  The  con- 
dition for  no  angular  acceleration  is  then  that  the  line  of  action 
of  the  force  (or  of  the  resultant  force  when  there  is  more  than 
one  force  acting)  pass  through  the  center  of  mass;  vice  versa,  when 
there  is  no  angular  acceleration  of  the  body,  the  line  of  action 
of  the  force  (or  of  the  resultant  force  when  there  is  more  than  one 
force  acting)  must  pass  through  the  center  of  mass. 

When  there  are  two  equal  and  opposite  external  forces  acting 
on  a  rigid  body,  there  can  be  no  linear  acceleration  of  the  body, 
i.e.,  no  acceleration  of  its  center  of  mass.  However,  when  the 
lines  of  action  of  these  two  forces  do  not  coincide,  the  moment  of 
the  two  forces  about  any  axis  will  not  be  equal,  hence  there  will  be 
an  angular  acceleration  of  the  body  about  its  center  of  mass.  The 
axis  of  this  acceleration  will  be  through  the  center  of  mass  per- 
pendicular to  the  plane  determined  by  the  line  of  action  of  the 
two  forces,  and  the  resultant  moment  of  the  two  forces  about  this 
axis  will  be  equal  to  the  product  of  either  force  by  the  perpendicu- 
lar distance  between  their  lines  of  action.  Two  such  equal  and 
opposite  forces  are  called  a  couple,  and  the  value  of  the  resultant 
moment  is  called  the  strength  of  the  couple.  Two  couples  balance 
each  other,  i.e.,  there  is  no  angular  acceleration,  when  their 
strengths  are  equal. 

21.  Work  and  Energy. — Whenever  one  portion  of  matter 
effects  a  change  in  some  other  portion  of  matter,  the  former  is 
said  to  do  work  on  the  latter.  The  attribute  or  condition  of 
matter  in  virtue  of  which  one  portion  of  matter  can  effect  changes 
in  other  portions  of  matter  is  called  energy;  that  is,  energy  is 
the  capacity  for  doing  work.  One  means  by  which  a  body  A 
may  produce  a  change  in  another  body  B  is  by  exerting  a  force  on 
the  latter  and  producing,  as  a  result  of  this  force,  a  displacement 


22  ELECTRICAL  ENGINEERING 

of  this  body  as  a  whole  or  a  displacement  of  its  individual  parts ; 
in  this  case  the  body  A  is  said  to  do  mechanical  work  on  the 
body  B.  As  the  measure  of  the  amount  of  mechanical  work 
done  on  a  particle  of  matter  by  the  force  which  causes  its  dis- 
placement is  taken  the  product  of  the  displacement  of  the  particle 
by  the  component  of  this  force  in  the  direction  of  the  displace- 
ment, provided  this  component  of  the  force  remains  constant 
during  the  displacement. 

In  general,  the  direction  of  the  displacement  as  well  as  the 
amount  and  the  direction  of  the  force  will  change  as  the  position 
of  the  particle  is  changed;  in  such  a  case,  the  above  definition 
applies  only  to  an  infinitesimal  displacement  of  the  particle,  i.e.,  to 
a  displacement  so  small  that  while  the  particle  is  being  displaced 
this  small  amount,  the  force  may  be  considered  constant  both  in 
amount  and  direction.  The  mechanical  work  W  done  on  a 
particle  displaced  any  finite  distance  /  is  then  the  sum  of  the 
products  of  the  force  for  each  infinitesimal  displacement  times 
the  component  of  this  displacement  in  the  direction  of  this  force ; 
i.e., 


W=         (fcosO)dl  (15) 

where/  represents  the  force  during  any  infinitesimal  displacement, 
dl  the  displacement  and  0  the  angle  between  the  direction  of  the 
displacement  and  the  direction  of  the  force. 

As  the  measure  of  the  amount  of  work  done  on  a  body  when 
a  change  is  produced  in  it  by  other  means  than  by  a  displacement 
of  the  body  as  a  whole  or  by  a  displacement  of  its  individual 
parts  under  the  action  of  a  force,  is  taken  the  amount  of  me- 
chanical work  which  would  be  required  to  produce  exactly  this 
same  change  were  this  change  effected  solely  by  a  displacement 
of  the  body  as  a  whole  or  by  a  displacement  of  its  individual  parts 
by  means  of  a  force  exerted  on  it.  For  example,  when  the  tem- 
perature of  a  body  is  increased  by  any  means  whatever,  the 
body  is  said  to  have  an  amount  of  work  done  on  it  equal  to  the 
amount  of  mechanical  work  which  would  have  to  be  done  on  it 
to  produce  exactly  this  same  rise  of  temperature. 

The  results  of  all  known  experiments  justify  the  assumption 
that  whenever  work  is  done  on  a  body  this  body  is  in  turn  given 
the  capacity  for  doing  an  exactly  equal  amount  of  work  on  other 
bodies,  arid  that  the  capacity  of  some  other  body  or  system  ot 


FUNDAMENTAL  IDEAS  AND   UNITS  23 

bodies  for  doing  work  is  diminished  by  an  exactly  equal  amount. 
That  is,  whenever  work  is  done,  some  body  or  system  of  bodies 
loses  an  amount  of  energy  equal  to  the  amount  of  work  done 
and  some  body  or  system  of  bodies  gains  an  exactly  equal  amount 
of  energy.  This  assumption,  which  is  justified  by  all  known 
experiments,  is  known  as  the  principle  of  the  conservation  of 
energy.  This  principle,  together  with  the  principles  of  the  Con- 
servation of  Mass,  the  Conservation  of  Linear  Momentum  and 
the  Conservation  of  Moment  of  Momentum,  are  the  four  cardinal 
principles  of  all  natural  science  and  engineering. 

It  is  found  convenient  in  discussing  the  various  properties 
of  matter  to  look  upon  each  property  of  a  portion  of  matter  as 
representing  a  definite  amount  of  energy  and  to  give  a  special 
name  to  the  energy  associated  with  each  property.  For  example, 
the  energy  associated  with  a  body  in  virtue  of  its  speed  is  called 
its  kinetic  energy;  the  energy  associated  with  a  body  in  virtue 
of  its  position  with  respect  to  other  bodies  which  exert  forces 
on  it  is  called  its  potential  energy;  the  energy  possessed  by  a 
body  in  virtue  of  its  temperature  is  called  its  heat  energy  or  its 
thermal  energy;  the  energy  associated  with  a  body  in  virtue  of 
its  chemical  nature  is  called  its  chemical  energy,  etc.  In  this 
terminology  the  principle  of  the  conservation  of  energy  may 
be  stated  thus:  The  only  possible  changes  which  can  take  place 
in  the  energy  associated  with  a  system  of  bodies  which  neither 
influences,  nor  is  influenced  by,  any  other  bodies,  are  changes 
in  form;  the  total  amount  of  energy  in  the  system  remains  unaltered. 

It  can  readily  be  shown  that  the  kinetic  energy  of  a  particle 
of  mass  m  moving  with  a  linear  speed  s  is  \  ms2.  A  rigid 
body  which  is  rotating  with  an  angular  speed  a>,  about  an  axis 
fixed  in  the  body  and  passing  through  its  center  of  mass,  which 
axis  at  the  same  time  has  a  linear  speed  s  (for  example,  the  arma- 
ture of  a  railway  motor  moving  relatively  to  the  earth)  has  a 
total  amount  of  kinetic  energy  equal  to 

iMs2  +  i/o>2  (16) 

relative  to  the  earth,  where  M  is  the  total  mass  of  the  body  and 
/  is  the  moment  of  inertia  of  the  body  about  the  axis  of  rotation 
fixed  in  the  body.  The  expressions  for  other  forms  of  energy 
will  be  given  when  the  corresponding  properties  of  matter  are 
discussed. 


24  ELECTRICAL   ENGINEERING 

The  unit  of  work  or  energy  on  the  c.  g.  s.  system  is  the  work 
done  when  a  force  of  one  dyne  displaces  a  particle  a  distance  of 
one  centimeter;  this  unit  is  called  the  erg.  Other  common  units 
of  work  and  energy  are  related  to  one  another  and  to  the  erg  as 
follows  : 

1  gram-centimeter  =980.5966  ergs   (at  sea  level 

and  45°  lat.) 

1  joule  =1  watt-second 

1  joule  =107  ergs 

1  kilogram-meter  =105  gram-centimeters 

1  foot-pound  =1.35573  joules 

1  foot-pound  =0.138255  kilogram-meter 

1  small  calorie  =0.001  large  calorie 

1  small  calorie  =4.186  joules 

1  small  calorie  =3.088  foot-pounds 

1  British  thermal  unit  =251.996  small  calories 

1  British  thermal  unit  =1,054.9  joules 

1  British  thermal  unit  =778.1  foot-pounds 

1  kilowatt-hour  =3,600,000  joules 

1  kilowatt-hour  =2,655,400  foot-pounds 

1  horsepower-hour  =2,684,300  joules 

1  horsepower-hour  =1,980,000  foot-pounds 

22.  Power.  —  Power  is  denned  as  the  time  rate  of  doing  work, 
or  as  the  time  rate  of  change  of  energy;  the  two  definitions  are 
equivalent. 

When  an  amount  of  energy  dW  is  transferred  in  an  infinitesimal 
interval  of  time  dt,  then  the  corresponding  power  is 


. 

dt  (17) 

When  this  rate  of  transfer  of  energy  is  constant,  then  the  power 
may  be  defined  as  the  amount  of  energy  transferred,  or  the 
work  done,  in  unit  time.  It  should  also  be  noted  that  since  energy 
or  work  is  the  product  of  force  and  displacement,  power  may  also 
be  defined  as  the  product  of  force  and  velocity. 

In  the  c.  g.  s.  system  the  unit  of  power  is  the  power  required 
to  do  work  at  the  rate  of  one  erg  per  second.  This  unit  is  seldom 
used  as  it  is  an  extremely  small  quantity;  instead  is  employed 
a  unit  which  is  10,000,000  ergs  per  second,  called  the  watt;  one 
watt  is  therefore  equal  to  one  j  oule  per  second.  Other  common  units 


FUNDAMENTAL   IDEAS  AND   UNITS 


25 


of  power  are  related  to  one  another  and  to  the  watt  as  follows : 


1  kilowatt 

1  megawatt 

1  metric  horsepower 

1  metric  horsepower 

1  horsepower 

1  horsepower 


=  1000  watts 
=  1000  kilowatts 
=75  kilogram-meters  per  second 
=0.735448  kilowatt 
=33,000  foot-pounds  per  minute 
=0.74565  kilowatt 
It  is  to  be  noted  that  torque  multiplied  by  angular  velocity 
(in  radians  per  unit  time)  gives  the  power  of  a  rotating  body  ex- 
pressed in  the  corresponding  units.     In  stating  the  performance 
of  electric  machines  use  is  frequently  made  of  the  expression 
"  torque  at  one-foot  radius  ";  this  is  equivalent  to  expressing  the 
torque  in  pound-feet.     (In  the  expression  "  torque  at  one-foot 
radius,"  the  word  torque  is  incorrectly  used,  since  torque  is  in- 
dependent of  the  radius  —  what  is  meant  is  the  "  force  "  at  one 
foot  radius.)     The  following  are  useful  relations : 

Power  in  kilowatts  =  1. 41 97  XlO"4X  torque    in    pound- 

feet  X  revolutions  per  minute. 

Power  in  horsepower  =  1 . 9040  X 1 0"4  X  torque    in    pound- 

feet  X  revolutions  per  minute. 
23.  Harmonic  Motion. — A  particular  type 
of  motion  which  is  not  only  of  considerable 
importance  itself,  but  also  serves  as  a  useful 
analogue  in  the  discussion  of  electric  circuits, 
is  that  known  as  harmonic  motion.  Harmonic 
motion  is  defined  as  motion  such  that  the 
acceleration  of  the  moving  body  is  propor- 
tional to,  and  in  the  opposite  direction  to, 
the  displacement  of  the  body  from  its  position 
of  equilibrium.  (The  equilibrium  position  of  a 
body  is  that  position  in  which  the  resultant 
force  acting  on  the  body  is  zero.)  Harmonic 
motion  is  illustrated  by  the  motion  of  a  pen- 
dulum, the  motion  of  a  weight  attached  to  a 
spring,  the  motion  of  each  particle  of  a  tuning 
fork,  the  motion  of  the  balance  wheel  of  a 
watch,  etc.  The  first  three  are  examples  of  harmonic  motion  of 
translation,  the  latter  of  harmonic  motion  of  rotation. 

Consider  the  case  of  a  simple  pendulum.     Let  B  be  the  bob 
of  the  pendulum  and  let  it  be  so  small  that  it  may  be  considered 


Fig.  8. 


26  ELECTRICAL  ENGINEERING 

as  a  single  particle  of  mass  m,  and  let  this  bob  be  suspended  by 
a  weightless  thread  of  length  I  from  a  fixed  point  P.  Let  the 
bob  be  displaced  from  its.  position  of  equilibrium  and  at  any 
time  t  let  its  displacement,  measured  along  the  arc  in  which  it 
moves,  be  x  =  arc  OB.  The  linear  velocity  at  which  the  bob 

dx 
is  moving  at  any  instant  is  then  —  and  its  kinetic  energy  is 

therefore  /  dx\2 

im  I  —  I 

\dt) 

The  potential  energy  of  the  bob  at  this  instant  is 
mg  0  Q  =mgl  (I  —  cos  ft) 

/>• 

where  ft  =—  and  g  is  the  acceleration  due  to  gravity.  The  total 
energy  of  the  bob  is  then 

Tf=i  m  (—\\mgl  (l-cos  ft) 
\dt  / 

and  if  there  is  no  friction  this  energy  must  be  constant,  and  there- 
fore the  rate  of  change  of  this  total  energy  with  respect  to  time 
must  be  zero.  Hence 

dW          dx    d*x  ,   dfi 


dft       1   dx 

whence,  since  —  =  —  —  —  , 
dt        I    dt 


_ 

For  ft  small,  sin  ft  =  ft  when  ft  is  expressed  in  radians  ;  whence 

T°  /7* 

for  —  small,  sin  B  =  —  and  therefore 

i  I 

<Px_     g 

*    T  (18) 

Equation  (18)*  tells  us  that  the  linear  acceleration  of  the  bob 
along  its  path  of  motion  is  proportional  to,  and  in  the  opposite 
direction  to,  its  displacement;  hence  this  is  a  case  of  harmonic 
motion. 

*This  equation  may  also  be  deduced  directly  from  a  consideration  of  the 
forces  acting  on  the  bob  at  any  instant. 


FUNDAMENTAL  IDEAS  AND  UNITS  27 

The  solution  of  any  differential  equation  of  the  form 

d*x 

~dt^-°)X 

where  CD  is  a  real  constant,  is 

x=X08in(o»t  +  0)  (19) 

where  X0  is  the  maximum  value  of  x,  and  8  is  a  constant  angle 
determined  by  the  value  of  x  for  t=0  and  the  maximum  value 
X0  of  the  variable  x.  Let  x0  be  the  value  of  x  for  t  =0,  then 


(20) 

This  angle  9  is  called  the  phase  angle  of  x,  and  measures  the  degree 
of  "  fullness  "  of  x  when  t=Q,  just  as  the  phase  of  the  moon 
represents  its  degree  of  fullness.  When  0=0,  x  starts  off  with 

TT 

its  zero  value  and  increases  ;  when  6  =  —  ,  x  starts  off  with  its 

maximum  or  full  value  and  decreases.  The  constant  o>  depends 
upon  the  value  of  t  required  for  x  to  pass  through  all  possible 
values  which  it  may  take  and  back  again  to  its  original  value; 
the  amount  by  which  t  must  increase  in  order  that  x  may  pass 
through  all  its  possible  values  and  back  again  is 


This  value  T  is  called  the  period  of  x.  The  number  of  periods 
corresponding  to  an  increase  of  unity  in  t  (e.g.,  if  t  represents 
time  in  seconds,  the  number  of  periods  per  second)  is  called  the 
frequency  of  x;  that  is,  the  frequency  is 

/=^or«  =  2,r/.  (22) 

Equation  (19)  may  be  represented  by  a  sine  curve,  Fig.  9, 
where  abscissas  are  o)  t  and  ordinates  x;  the  distance  along  the 
axis  of  ait  between  the  points  where  the  curve  cuts  this  axis  in 
the  same  direction  is  equal  to  2ir  radians  or  360  degrees.  The 

curve  marked  "x"  in  the  figure  is  plotted  for  the  case  when  9  =— 
that  is,  for  x  a  maximum  and  decreasing  at  o>£=0.  The  curve 


28 


ELECTRICAL  ENGINEERING 


marked  "  v  "  is  for  6  =  TT.  Note  that  the  angle  6  in  any  case 
is  also  equal  to  the  distance  to  the  left  of  the  origin  at  which 
the  curve  first  crosses  the  base  line  in  the  positive  direction. 
A  function  of  the  form  X0  sin  (a)t  +  0},  that  is,  a  function  which 
can  be  represented  by  a  single  sine  wave,  is  sometimes  called 


\  \ 


Fig.  9. 

a  simple  harmonic  function,  or  briefly,  a  harmonic  function.     Other 
periodic  functions  are  called  non-harmonic  functions. 

In  the  case  of  the  simple  pendulum  let  the  bob  be  held  out 
a  distance  A  from  its  equilibrium  position,  and  at  a  given  instant 
let  it  be  released.  Let  time  be  counted  from  this  instant;  then 
for  t  =0  we  have 

35=4 

and  the  velocity  at  this  instant  is 

dx 


Substitution  of  these  values  in  (19)  gives 

A  =X0  sin  0 
0=XQ  cos  9 

whence  9=—    and  X0=A.     Therefore    the  displacement  of  the 
tt 

bob  from  its  equilibrium  position  at  any  instant  is 
x  =  A  sin  (  a)  £  +  —  \  =A  cos  a)  t 

and  its  velocity  at  this  instant  is 

dx 

v=  —   =  —  ft)  A  sin  CD  t 
dt 


FUNDAMENTAL  IDEAS  AND  UNITS  29 

where  G>=-*ly.  Note  that  the  displacement  has  its  first  positive 
maximum  after  the  start  for  a)  t  =  2  TT  and  the  velocity  its  first 
positive  maximum  for  o>  t  —  —  ;  hence  the  velocity  reaches  a  pos- 

itive maximum  a  quarter  of  a  period  ahead  of  the  displacement,  or 
the  velocity  leads  the  displacement  by  90°  (see  Fig.  9).  Two  sine 
waves  of  the  same  frequency  which  reach  their  positive  maxima 
at  different  times  are  said  to  differ  in  phase  by  the  angle  corre- 
sponding to  the  time  interval  between  successive  positive  maxima 
of  the  two  waves.  When  both  waves  are  expressed  as  sine 
functions,  the  difference  in  phase  is  the  difference  between  the 
phase  angles  of  the  two  functions.  In  the  above  example 


la)t  +  -~\ 


x=A  sin 
v=a)A  sin 

whence  the  phase  angle  of  x  is  —  and  the  phase  angle  of  v  is  TT; 

2i 

therefore  the  difference  in  phase  between  the  two  is 

77        7T 

TT—  -  =  —     or  90  degrees. 

The  function  with  the  larger  (algebraically)  phase  angle  always 
leads.  Note  that  the  leading  curve  is  to  the  left. 

24.  Temperature.  —  The  physical  properties  of  any  piece  of 
matter  depend,  among  other  things,  upon  its  temperature,  i.e., 
upon  its  relative  hotness  or  coolness  referred  to  some  standard  sub- 
stance under  standard  conditions.  The  idea  of  temperature  is 
familiar  to  every  one,  and  one's  so-called  temperature  sense  en- 
ables one  to  form  a  rough  judgment  of  the  relative  hotness  or  cool- 
ness of  two  or  more  bodies.  For  scientific  purposes,  however,  a 
more  reliable  and  more  delicate  means  of  "  measuring  "  tem- 
perature is  desirable.  Any  device  which  serves  this  purpose  is 
called  a  thermometer.* 

The  standard  temperature-measuring  device  is  the  constant 
volume  hydrogen  thermometer,  which  consists  essentially  of  a 
suitable  receptacle  containing  a  constant  mass  of  hydrogen  gas 
kept  at  constant  volume,  with  means  provided  for  measuring  any 

*A  thermometer  designed  to  measure  very  high  temperatures  is  called  a 
"pyrometer." 


30  ELECTRICAL  ENGINEERING 

variation  that  may  be  caused  to  take  place  in  the  pressure  of  the 
gas.  The  numerical  value  of  the  temperature  of  any  substance 
is  then  defined  in  terms  of  the  relative  pressure  of  this  gas  when  the 
receptacle  is  immersed  in  the  substance,  referred  to  the  pressure 
of  this  gas  when  the  receptacle  is  immersed  in  melting  ice  at  a  pres- 
sure of  760  mm.  of  mercury,  the  pressure  of  the  gas  in  each  case 
being  measured  after  it  has  reached  a  constant  value.  The  tem- 
perature of  melting  ice  at  a  pressure  of  760  mm.  of  mercury  is 
arbitrarily  taken  as  zero  degrees,  and  the  temperature  of  saturated 
steam  at  a  pressure  of  760  mm.  of  mercury  is  taken  as  100  degrees. 
Calling  P!  the  pressure  of  the  hydrogen  gas  when  the  receptacle  is 
immersed  in  the  melting  ice  and  p2  its  pressure  when  immersed 
in  the  saturated  steam,  and  p  its  pressure  when  immersed  in  any 
given  substance  S  (the  pressure  in  each  case  being  measured  after 
the  lapse  of  a  sufficient  time  for  it  to  reach  a  constant  value),  the 
numerical  value  of  the  temperature  of  the  given  substance  is 

7) —  P, 

defined  as  t  =  - — — XlOO  degrees  centigrade. 

Prpi 

The  Fahrenheit  scale  of  temperature  is  derived  in  the  same 
manner,  except  that  the  temperature  of  the  melting  ice  is  taken  as 
32  degrees  and  that  of  the  saturated  steam  as  212  degrees.  A 
temperature  of  tf  degrees  Fahrenheit  is  then  equal  to 

tc  =f-  (tt  -  32)  degrees  centigrade.  (23) 

For  practical  purposes,  the  volume  expansion  of  mercury  in 
glass,  or  of  alcohol  in  glass,  is  used  as  a  measure  of  temperature. 
For  strictly  accurate  measurements,  such  mercury-in-glass  or 
alcohol-in-glass  thermometers  should  be  calibrated  by  comparison 
with  a  standard  hydrogen  thermometer,  but  for  ordinary  work  it 
is  sufficient  to  determine  the  two  points  on  the  thermometer  scale 
corresponding  to  the  temperature  of  melting  ice  and  saturated 
steam  under  standard  conditions,  and  to  assume  that  the  volume 
expansion  of  the  thermometer  fluid  is  proportional  to  the  tem- 
perature. 

25.  Heat  Energy.  —  All  known  experiments  lead  us  to  believe 
that  whenever  the  temperature  of  a  body  increases,  energy  is 
transferred  to  it,  and  that  whenever  the  temperature  of  a  body 
decreases  energy  is  transferred  from  it;  further,  that  there  is 
a  fixed  numerical  relation  between  the  quantity  of  energy  trans- 
ferred to  a  given  body  and  its  change  in  temperature.  This  nu- 


FUNDAMENTAL  IDEAS  AND  UNITS  31 

merical  relation  can  be  determined  directly  only  in  case  the 
change  in  temperature  is  produced  by  mechanical  work,  since, 
by  definition,  the  measure  of  energy  is  the  product  of  a  force 
by  a  distance.  Careful  experiments  have  shown  that  the  work 
required  to  raise  the  temperature  of  one  gram  of  water  from 
0°  to  100°  centigrade  is  418.6  XlO7  ergs.  We  may  then  take  as 
the  unit  of  heat  energy  the  one-hundredth  part  of  the  work 
required  to  raise  one  gram  of  water  from  0  to  100  degrees  centi- 
grade. This  unit  is  known  as  the  mean  small  calorie,  and  in 
terms  of  it,  the  amount  of  energy  involved  in  various  heat  effects 
may  be  measured  with  comparative  ease.  It  should  be  noted 
that  the  work  required  to  raise  one  gram  of  water  one  degree 
is  different  at  different  temperatures;  the  variation  is  negligible, 
however,  except  in  the  most  refined  work. 

The  above  numerical  relation  between  the  mean  small  calorie 
and  the  erg  is  called  the  mechanical  equivalent  of  heat  on  the  c.  g.  s. 
system.  Another  way  of  expressing  the  mechanical  equivalent 
of  heat  is,  the  number  of  foot-pounds  of  work  required  to  raise 
the  temperature  of  one  pound  of  water  one  degree  Fahrenheit 
or  at  near  its  maximum  density  (39.1°  Fahrenheit) ;  this  may 
likewise  be  used  as  a  unit  of  heat  energy,  and  is  known  as  the 
"  British  Thermal  Unit."  See  Article  21  for  the  relations  between 
the  various  units  of  heat  energy. 

Practically  every  phenomenon  in  nature  is  accompanied  by  a 
change  in  temperature  of  one  or  more  bodies.  In  certain  cases 
the  entire  amount  of  energy  transferred  in  the  process  may  be 
caused  to  appear  as  heat  energy,  and  when  this  can  be  done  the 
total  amount  of  energy  transferred  can  be  measured  with  a  fair 
degree  of  accuracy.  Usually,  the  heat  energy  developed  in  any 
process  is  not  in  a  useful  form;  in  such  cases  the  heat  energy 
is  said  to  be  "  dissipated." 

26.  Efficiency  and  Losses.  —  In  any  machine  or  apparatus 
which  is  employed  for  transforming  energy  from  one  form  into 
another  or  for  transferring  energy  from  one  place  to  another, 
a  certain  amount  of  energy  is  always  converted  into  forms  which 
cannot  be  readily  utilized.  In  general,  this  useless  energy  appears 
as  heat  energy.  The  rate  at  which  energy  is  put  into  a  machine 
is  called  the  power  input  into  the  machine  and  the  corresponding 
rate  at  which  the  machine  gives  out  useful  energy  is  called  the 
power  output,  or  the  load  on  the  machine.  The  difference  between 


32  ELECTRICAL    ENGINEERING 

the  power  input  and  the  power  output  is  called  the  power  loss. 
The  ratio  of  the  output  P0  to  the  input  Pt  for  any  given  output 
of  a  machine  is  denned  as  the  efficiency  of  the  machine  at  this 
output;  this  ratio  is  usually  expressed  as  a  percentage.  The 
per  cent  efficiency  e  is  then 

e  =  100^°.  (24) 

*  i 

The  ratio  of  the  difference  between  the  input  Pi  and  the  output 
P0  to  the  input  Pi  expressed  as  a  percentage,  is  called  the  per  cent 
power  loss  p;  that  is, 

p  __  p 
p  =  100  —  —  -  =  100-e.  (25) 

P.i 

Sometimes  it  is  more  convenient  to  express  the  power  loss  as  a 
percentage  of  the  output  P0.  Let  p'  be  the  percentage  loss 
expressed  in  this  manner  ;  then 

(26) 


- 
P0         100  -p 

The  efficiency  of  a  machine  varies  as  a  rule  with  the  load 
or  output..  For  no  load,  i.e.,  no  output,  the  efficiency  is  zero, 
since  in  general  energy  must  be  supplied  to  the  machine  to  operate 
it,  even  though  the  machine  does  no  useful  work;  as  the  load 
comes  on  the  efficiency  increases  up  to  a  certain  output,  depend- 
ing on  the  design  of  the  machine,  and  then  decreases. 

The  rated  load  or  rated  power  output  of  a  machine  which 
is  designed  for  continuous  service  is  the  maximum  rate  at  which 
useful  energy  may  be  transferred  through  the  machine  con- 
tinuously without  injury  to  any  of  its  parts.  In  most  electric 
machinery  this  is  determined  by  the  rise  of  temperature  pro- 
duced by  the  energy  dissipated  or  "  lost  "  in  the  windings  and 
iron  cores.  As  a  rule,  the  insulation  of  the  windings  will  deterio- 
rate if  the  temperature  of  the  machine  exceeds  75°  centigrade. 

27.  Newton's  Law  of  Gravitation.  —  From  the  observations  by 
various  astronomers  of  the  revolution  of  the  planets  about  the  sun, 
and  of  the  revolution  of  the  moon  about  the  earth,  Newton  was 
led  to  the  belief  that  every  particle  of  matter  in  the  universe 
attracts  every  other  particle  with  a  force  proportional  to  the  pro- 
duct of  the  masses  of  the  two  particles,  and  inversely  proportional 
to  the  square  of  the  distance  between  them,  namely  with  a  force 


'  •     T  (27) 


FUNDAMENTAL  IDEAS  AND  UNITS  33 

where  &  is  a  constant  depending  upon  the  units  in  which  m,  m',  r, 
and  /  are  measured.  Further  observations  and  careful  laboratory 
experiments  by  Cavendish  and  others  have  confirmed  this  belief 
which  is  now  accepted  as  one  of  the  fundamental  "  laws  "  of 
nature.  The  constant  k  has  been  determined  by  experiment  to  be 
6.66 XlO"8  when  m  and  m'  are  expressed  in  grams,  r  in  centi- 
meters, and  /  in  dynes.  For  small  masses,  therefore,  such  as  one 
ordinarily  deals  with  in  the  laboratory,  this  force  is  extremely 
small,  and  can  be  detected  only  with  the  most  delicate  instruments. 
As  we  shall  see  presently,  the  mutual  attraction  or  repulsion 
between  magnetic  poles  and  also  the  mutual  attraction  or  repul- 
sion between  electric  charges  obey  a  law  of  exactly  the  same  form. 
Hence  the  study  of  forces  which  obey  this  law  becomes  of  prime 
importance.  It  should  be  borne  in  mind  that  any  conclusions 
derived  from  this  law  of  force  action  will  apply  to  any  one  of  the 
three  agents,  gravitational  masses,  magnetic  poles,  and  electric 
charges.  In  order  to  express  such  deductions  in  terms  of  physical 
quantities,  we  shall  consider  first  the  forces  produced  by  magnetic 
poles. 

PROBLEMS 

1 .  Find  the  speed  in  meters  per  second  and  in  feet  per  second 
of  an  electric  locomotive  travelling  at  a  uniform  speed  of  60  miles 
per  hour.     If  the  locomotive  is  travelling  at  this  speed  around 
a  curve  of  1000  feet  radius,  what  is  the  direction  and  amount 
of  the  acceleration  in  feet  per  second  per  second  and  in  miles 
per  hour  per  second? 

Ans.:  26.8  meters  per  second;  88.0  feet  per  second;  7.74  feet 
per  second  per  second ;  5.28  miles  per  hour  per  second.  Accelera- 
tion toward  center  of  curvature. 

2.  In  Problem  1,  if  the  distance  between  rails  is  4  feet,  8.5 
inches,  how  many  inches  would  the  outer  rail  have  to  be  elevated 
above  the  inner  rail  in  order  that  the  flanges  of  the  locomotive 
wheels  exert  no  side  thrust  on  the  rails? 

Ans.:  13.60  inches.  (In  practice  the  elevation  would  be 
about  half  this ;  that  is,  the  actual  force  on  the  outer  rail  would 
be  about  half  what  it  would  be  were  both  rails  in  the  same  hori- 
zontal plane.) 

3.  If  in  Problem  1  the  outer  rail  of  the  curve  is  elevated  6 
inches  and  the  locomotive  weighs   100  tons,  what  will  be  the 


34  ELECTRICAL    ENGINEERING 

average  shearing  force  exerted  on  each  of  the  spikes  holding 
the  rails  to  the  ties?     Assume  the  total  force  to  be  exerted  against 
15  spikes,  and  neglect  the  friction  of  the  rail  against  the  ties. 
Gauge  of  track  4  feet,  8.5  inches. 
Ans.:  1794  pounds. 

4.  A  man  stands  on  the  floor  of  a  trolley  car  which  is  travelling 
at  the  rate  of  20  miles  an  hour  around  a  curve  of  100  feet  radius. 
How  many  degrees  from  the  vertical  must  the  man  lean  to  prevent 
his  falling   over?     Assume  the  man  may  be  represented  by  a 
rod  standing  on  its  end.     Draw  a  complete  vector  diagram  of 
the  forces  acting. 

Ans.:  15.0.° 

5.  The  tangential  force  exerted  by  a  brake  on  a  pulley  which 
is  rotating  at  a  speed  of  500  revolutions  per  minute  is  200  pounds. 
If  the  diameter  of  the  pulley  is  one  foot,  calculate  the  torque 
in  pound-feet  developed  by  the  pulley,  and  the  amount  of  work 
in  horsepower-hours  done  by  the  pulley  in  10  minutes. 

Ans.:  100  pound-feet;  1.587  horsepower-hours. 

6.  The  radius  of  gyration  of  a  cylinder  about  its  own  axis 

D 

is  equal  to  —  where  D  is  the  diameter  of  the  cylinder.     What  is 

the  moment  of  inertia  of  an  iron  cylinder  5  feet  in  diameter  and 
4  feet  long?     Specific  gravity  of  iron  7.7 ;  give  answer  in  pound- 
foot  units.  _2 
Ans.:  59,100  pound-foot. 

7.  If  the  cylinder  in  Problem  6  is  rotating  at  a  speed  of  1000 
revolutions  per  minute  what  is  its  kinetic  energy  in  foot-pounds 
and  in  horsepower- hours? 

Ans.:  324,000,000  foot-pounds;  163.6  horsepower-hours. 

8.  If  the  force  driving  the  cylinder  in  Problem  7  is  removed, 
how  many  hours  will  it  take  the  cylinder  to  come  to  rest  if  the 
opposing   torque  due  to  friction  is   constant   and   equal  to  300 
pound-feet? 

Ans.:  5.73  hours. 

Note :  The  data  given  in  Problems  6  to  8  are  fairly  representa- 
tive of  a  large  turbo-generator,  except  that  the  friction  torque 
is  not  constant.  Unless  a  brake  of  some  kind  is  used  the  rotating 
part  of  such  a  machine  will  continue  in  motion  for  several 
hours  after  the  steam  is  shut  off. 

9.  A  weight  of  500  pounds  falls  from  rest  a  distance  of  100 


FUNDAMENTAL  IDEAS  AND  UNITS          35 

feet  into  a  tub  of  water.  If  the  tub  contains  10  gallons  of  water, 
and  all  the  kinetic  energy  lost  by  the  weight  when  it  is  brought 
to  rest  goes  into  heating  the  water,  what  will  be  the  rise  in  the 
temperature  of  the  water  produced  thereby,  assuming  no  radia- 
tion of  heat  energy?  Give  answer  in  degrees  centigrade  and 
degrees  Fahrenheit. 

Ans.:  0.430°  centigrade;  0.774°  Fahrenheit. 

10.  The  thermal  efficiency  of  the  best  modern  boiler  is  about 
65  per  cent ;  i.e.,  of  the  total  energy  in  the  coal  65  per  cent  can 
be  transferred  to  the  steam.     The  efficiency  of  a  modern  steam 
engine  working  under  the  best  conditions  is  about  20  per  cent; 
i.e.,  of  the  total  energy  in  the  steam  which  passes  through  the 
engine  only  20  per  cent  is  converted  into  mechanical  energy. 
A  pound  of  high-grade  coal  contains  14,000  British  thermal  units 
of  energy.     Assuming  a  boiler  efficiency  of  65  per  cent  and  an 
engine  efficiency  of  20  per  cent,  how  many  pounds  of  coal  will 
be  required  to  produce  one  horsepower-hour  at  the  engine  shaft? 
What  is  the  over  all  efficiency  of  boiler  and  engine? 

Ans.:  1.40  pounds  of  coal  per  horsepower-hour;  13  per  cent. 
(In  practice  a  considerably  greater  amount  of  coal  is  required 
per  horsepower-hour,  due  to  the  fact  that  an  engine  seldom 
works  at  full  load  all  day  and  the  efficiency  is  less  for  light  loads. 
A  large  steam  plant  developing  from  10,000  to  40,000  horsepower 
maximum  load  uses  about  2  pounds  of  coal  per  horsepower  hour; 
small  plants  of  a  few  hundred  horsepower  capacity  use  about  5 
pounds  of  coal  per  horsepower  hour.) 

11.  If  the  efficiency  of  a  water  wheel  is  80  per  cent,  how  many 
cubic  feet  of  water  per  second  falling  through  a  distance  of  one 
foot  will  be  required  to  develop  energy  at  the  rate  of  one  horse- 
power?    Let  Q  be  the  number  of  cubic  feet  per  second,   H  the 
"  head  "  or  distance  through  which  the  water  falls,  and  P  the 
horsepower;  what  is  the  relation  between   P,    Q,  and    //  for  a 
wheel  of  80  per  cent  efficiency? 

QH 
Ans.:  11  cubic  feet  per  second;  P=  —• 

12.  A  lead  ball  which  has  a  mass  of  1  pound  is  suspended 
by  a  light  string  from  a  fixed  support;  distance  from  center  of 
mass  of  ball  to  support  1  yard.     The  ball  is  displaced  a  horizontal 
distance  of  3  inches  from  the  vertical  and  then  let  go.     What  is 
the  expression  for  the  displacement  (in  inches)  of  the  ball  from  the 


36  ELECTRICAL  ENGINEERING 

vertical  t  seconds  later,  neglecting  the  effect  of  friction ;  what  will 
be  its  maximum  velocity  in  inches  per  second  and  what  will  be 
its  displacement  from  the  vertical  at  this  instant;  what  will 
be  the  frequency  of  the  pendulum? 

Ans.:  x=3  cos  (3.270  inches;  9.81  inches- per  second  every 
time  the  displacement  from  the  vertical  is  zero;  0.52  cycles  per 
second. 

13.  Plot  to  scale  the  values  of  the   displacement,   velocity, 
and   acceleration  of   the   vibrating   bob   in   Problem   12.     What 
are  the  phase  relations  between  the  velocity  and  the  displace- 
ment; between  the  velocity  and  the  acceleration? 

Ans.:  The  velocity  leads  the  displacement  by  90°  and  lags 
90°  behind  the  acceleration. 

14.  What  will  be  the  kinetic  energy  of  the  ball  in  Problem  12 
when  it  is  half  way  between  its  equilibrium  position  and  its  point 
of  maximum  deflection?     What  will  be  its  potential  energy  at 
this  instant?     Give  answers  in  joules  and  in  foot-pounds. 

Ans.:  Kinetic  energy  0.251  foot-pounds  or  0.340  joules; 
potential  energy  0.084  foot-pounds  or  0.114  joules. 

15.  Two  lead  spheres  each  1  foot  in  diameter  are  placed  near 
each  other  with  their  nearest  points  1  inch  apart.     The  density 
of  lead  is  11.3.     Calculate  the  force  (gravitational)  in  dynes  with 
which  the  two  spheres  attract  each  other,   assuming  the  total 
mass  of  each  sphere  concentrated  at  its  center. 

Ans.:  1.73  dynes. 


II 

MAGNETISM 

28.  Magnets.  —  Magnetic  Poles.  —  It  has  long  been  known 
that  a  certain  mineral,  called  the  loadstone,  has  the  property  of 
attracting  pieces  of  iron  or  steel  with  a  readily  perceptible  force, 
even  though  the  loadstone  and  the  piece  of  iron  or  steel  be  com- 
paratively small.     In  other  words,  between  loadstone  and  iron  or 
steel  there  is  a  force  of  attraction  many  times  greater  than  the 
force  of  attraction  between  like  masses  of  ordinary  matter.     It 
has  also  been  known  for  many  centuries  that  a  steel  bar  can  be 
given  this  same  property  by  stroking  it  lengthwise  with  a  piece  of 
loadstone,  and  that  when  the  bar  thus  treated  is  freely  suspended, 
it  takes  up  a  definite  position  with  respect  to  the  earth,   one  end 
of  the  bar  pointing  approximately  toward  the  north  geographical 
pole  and  the  other  end  toward  the  south  geographical  pole.     A 
steel  bar  having  this  property  is  called  a  magnet,  the  north  pointing 
end  is  called  its  north  or  positive  pole  and  the  south  pointing  end  its 
south  or  negative  pole.     In  general,  a  magnet  may  be  defined  as  a 
body  which  possesses  the  property  of  attracting  with  a  readily  per- 
ceptible force  pieces  of  iron  or  steel,  and  which,  when  freely  sus- 
pended, takes  up  a  definite  position  with  respect  to  the  geographical 
meridian.     As  we  shall  see  later  on,  it  is  possible  to  "  magnetise  " 
a  steel  bar  in  other  ways  than  by  stroking  it  with  a  piece  of  load- 
stone; in  particular,  when  an  insulated  wire  is  wrapped  around 
such  a  bar  and  an  electric  current  is  established  in  the  wire,  the 
bar  becomes  magnetised;  this  is  the  modern  way  of  making  a 
magnet  and  the  only  way  of  making  a  powerful  one. 

29.  Paramagnetic  and  Diamagnetic  Substances.  —  It  is  found 
by  experiment  that  a  magnet  attracts  not  only  iron  and  steel,  but 
to  a  less  extent  nickel  and  cobalt.     Bismuth,  on  the  other  hand,  is 
repelled  by  a  magnet.   Substances  which  of  themselves  are  not  mag- 
nets but  which  are  attracted  by  a  magnet  are  called  paramagnetic 
substances,  or  simply  magnetic  substances;  substances  which  of 
themselves  are  not  magnets  but  which  are  repelled  by  a  magnet  are 
called  diamagnetic  substances.     Except  in  the  case  of  iron,  steel, 

37 


38  ELECTRICAL    ENGINEERING 

nickel,  and  cobalt,  the  force  of  attraction  for  paramagnetic  bodies 
is  extremely  small;  of  the  diamagnetic  substances  bismuth  is  the 
only  one  which  is  strongly  repelled  by  a  magnet,  and  the  repul- 
sion even  of  bismuth  is  weak  compared  to  the  attraction  of  iron. 

Whether  or  not  a  substance  is  attracted  or  repelled  by  a 
magnet  depends  upon  the  nature  of  the  medium  surrounding  the 
magnet  and  the  substance.  For  example,  it  is  found  by  experi- 
ment that  a  solution  of  perchloride  of  iron  is  attracted  by  a  magnet 
when  the  solution  and  the  magnet  are  surrounded  by  air.  On  the 
other  hand,  if  the  magnet  is  immersed  in  the  solution  and  a  bubble 
of  air  formed  in  the  latter,  the  air  will  be  repelled  by  the  magnet. 
Therefore  perchloride  of  iron  is  paramagnetic  with  respect  to  air,, 
and  air  is  diamagnetic  with  respect  to  perchloride  of  iron.  In 
other  words,  paramagnetic  and  diamagnetic  are  purely  relative 
terms.  It  is  customary  to  take  air  as  the  standard  of  reference, 
although  in  certain  cases  it  would  be  preferable  to  take  free  space 
or  the  "  ether  "  as  the  standard.  However,  the  difference  be- 
tween air  and  free  space  in  respect  to  their  magnetic  qualities  is 
practically  inappreciable,  so  that  a  body  which  is  magnetic  with 
respect  to  air  is  likewise  magnetic  with  respect  to  free  space.  In 
the  following  discussion,  when  the  terms  paramagnetic  and  dia- 
magnetic are  used,  air  is  to  be  understood  as  the  standard  of 
reference;  i.e.,  air  is  assumed  to  be  non-magnetic. 

30.  Attraction  and  Repulsion  of  Magnetic  Poles.  —  A  piece  of 
iron  or  other  magnetic  substance  which  of  itself  is  not  a  magnet, 
will  be  attracted  by  either  pole  of  a  magnet  when  placed  near  this 
pole.     However,  when  two  magnets  are  placed  near  each  other, 
the  mutual  force  between  the  two  may  be  either  an  attraction  or  a 
repulsion.     When  the  two  like  poles  of    the  two  magnets   are 
nearer  together  than  their  unlike  poles,  the  force  is  a  repulsion; 
when  the  unlike  poles  are  nearer  together,  the  force  is  an  attraction. 
This  is  readily  tested  by  suspending  one  of  the  magnets  by  a  thread 
attached  to  its  middle  point  and  bringing  the  two  ends  of  the 
second  magnet  successively  near  one  end  of  the  suspended  magnet. 
We  therefore  conclude  that  like  magnetic  poles  repel  each  other 
and  unlike  poles  attract  each  other.     Experiment  also  shows  that 
the  force  of  attraction  or  repulsion  of  two  magnets  falls  off  rapidly 
as  the  distance  between  them  is  increased. 

31.  Magnetic  Charge. —  A  Magnetic  Pole  as  a  Force-Producing 
Agent.  —  It  is  found  by  experiment  that  the  forces  produced  by 


MAGNETISM  39 

magnets  on  one  another  and  upon  magnetic  bodies  placed  in  their 
vicinity  may  be  accounted  for  in  terms  of  a  something  confined 
solely  to  the  external  surfaces  of  such  bodies.  In  the  case  of  a 
long,  slim  magnet  this  force-producing  agent  is  confined  almost 
entirely  to  the  ends  of  the  magnet.  Since  the  two  ends  of  a  mag- 
net possess  opposite  properties,  it  is  necessary  to  look  upon  this 
force-producing  something  as  different  at  the  two  ends  of  the 
magnet.  This  something  at  the  north  pointing  end  of  a  magnet 
may  be  called  a  "  positive  magnetic  charge/'  and  the  something 
at  the  south  pointing  end  a  "  negative  magnetic  charge."  How- 
ever, it  is  customary  to  use  the  expression  "  north  (or  positive) 
magnetic  pole  "  to  signify  this  something  at  the  north  pointing 
end  of  a  magnet  and  the  expression  "  south  (or  negative)  magnetic 
pole  "  to  signify  this  something  at  the  south  pointing  end.  When 
used  in  this  sense,  the  word  "  pole  "  signifies  the  force-producing 
agent  associated  with  the  surface  of  a  magnet  rather  than  the 
particular  part  of  the  magnet  at  which  this  force-producing  agent 
is  located. 

It  is  to  be  noted  that  although  the  external  forces  produced  by 
a  magnet  may  be  accounted  for  in  terms  of  a  something  confined 
solely  to  its  external  surface,  there  is  also  a  change  produced 
throughout  the  substance  of  a  body  when  it  is  magnetised.  For, 
when  such  a  body  is  broken  in  two,  the  two  broken  ends  are 
found  to  be  the  seat  of  equal  and  opposite  magnetic  poles. 
There  are  also  certain  very  special  cases  in  which  the  poles  of  a 
magnetised  body  must  be  considered  as  distributed  throughout 
its  interior,  but  the  theory  developed  below  for  surface  poles  may 
be  readily  extended  to  such  cases. 

32.  Induced  Magnetisation.  —  In  order  to  express  in  an  exact 
quantitative  manner  the  value  of  the  mutual  forces  produced  on 
one  another  by  magnetic  poles  it  is  always  necessary  to  take  into 
account  the  effect  of  any  magnetic  bodies  which  may  be  in  their 
vicinity.  As  we  have  seen,  when  a  magnetic  body  which  itself  is 
not  a  magnet  (a  small  rod  of  soft  iron,  for  example)  is  placed  in 
the  vicinity  of  a  magnetic  pole,  this  substance  is  attracted  whether 
the  pole  is  a  north  or  a  south  pole.  In  fact,  the  body  acts  exactly 
like  a  magnet,  except  that  the  strength  and  position  of  its  poles 
depend  on  its  position  with  respect  to  the  magnet  attracting  it. 
The  magnetic  body  is  therefore  said  to  be  magnetised  by  "  induc- 
tion." Since  the  magnetised  body  is  always  attracted,  the  loca- 


40  ELECTRICAL  ENGINEERING 

tion  of  the  poles  induced  on  it  is  always  such  that  two  of  the  unlike 
poles  on  the  attracting  magnet  and  on  the  magnetised  body  re- 
spectively are  nearer  together  than  the  like  poles  on  these  two 
bodies.  A  diamagnetic  body  placed  in  the  vicinity  of  a  magnet 
is  similarly  magnetised  by  induction,  except  that  in  this  case, 
since  there  is  a  repulsion  between  the  inducing  magnet  and  the 
diamagnetic  body,  the  like  poles  are  nearer  together.  For 
example,  a  small  rod  A  B  of  soft  iron  placed  in  the  vicinity  of  a 
magnet  in  the  various  positions  shown  in  Fig.  10,  is  magnetised  as 


\n_ 


Fig.   10. 

indicated.  When  the  rod  is  removed  from  the  vicinity  of  the 
magnet  the  induced  poles  disappear  almost  entirely.  In  case  a 
hard  steel  rod,  originally  unmagnetised,  is  placed  near  the  mag- 
net, it  becomes  likewise  magnetised  by  induction,  though  to  a 
less  extent  than  the  soft  iron.  If,  however,  the  steel  rod  is  removed 
from  the  vicinity  of  the  magnet  it  is  found  to  retain  to  a  consid- 
erable degree  its  magnetisation,  that  is,  the  rod  becomes  a  "  per- 
manent "  magnet. 

The  phenomenon  of  induced  magnetisation  is  a  most  important 
one  both  in  practice  and  in  theory,  and  we  shall  return  to  this  sub- 
ject for  a  more  detailed  study.  The  chief  fact  to  be  borne  in  mind 
for  the  present  is  that  any  magnetic  or  diamagnetic  body  placed  in 
the  vicinity  of  a  magnet  has  magnetic  poles,  induced  on  its  surface, 
and  that  these  induced  poles  produce  forces  on  every  magnetic  pole, 
permanent  or  induced,  which  may  be  in  the  vicinity.  For  example, 
when  two  magnets  are  separated  from  each  other  by  a  given  dis- 
tance and  there  are  no  magnetic  or  diamagnetic  bodies  in  the 
vicinity,  they  will  exert  a  certain  force  upon  each  other.  When  a 
magnetic  body,  such  as  a  piece  of  unmagnetised  soft  iron,  is  placed 


MAGNETISM 


41 


near  them  (see  Fig.   11),  the  force  acting  on  each  magnet  will 
evidently  be  changed,  since  the  poles  s  and  n  induced  on  the 


Fig.   11. 

soft  iron  also  exert  forces  upon  each  magnet.  In  the  case  illus- 
trated, the  resultant  force  acting  on  each  magnet  will  be  increased 
and  also  changed  in  direction.  Again,  when  the  entire  region 
around  the  two  magnets  is  filled  with  a  magnetic  liquid  (per- 
chloride  of  iron,  for  example),  magnetic  poles  are  induced  on 
the  surface  of  the  liquid  where  it  comes  in  contact  with  the  poles 
of  the  magnets,  Fig.  12,  and  these 
induced  poles  are  of  the  opposite 
sign  to  those  of  the  magnet  at 
this  surface.  The  effect  of  these 
induced  poles  in  this  case  is  to 
decrease  the  resultant  force  on 
each  magnet,  Art.  56.  The  as- 
sumption that  air  is  non-magnetic 
is  equivalent  to  assuming  that  there 
are  no  poles,  either  permanent 
or  induced,  produced  on  the  air 
in  contact  with  a  magnet.  Fis-  12- 

33.  Point-Poles.  —  Experiment  shows  that  it  is  physically  im- 
possible to  have  a  magnetic  "  charge  "  or  pole  of  finite  amount  con- 
centrated in  a  point.     However,  for  the  purposes  of  mathematical 
analysis  of  the  forces  produced  by  magnets,  it  is  frequently  con- 
venient to  consider  a  magnetic  pole  as  occupying  but  a  point  in 
space.     We    may   call    such   a   pole   a   point-pole.     A   physical 
approximation  to  such  a  point-pole  is  the  pole  on  a  very  small  area. 

34.  Properties  of  Magnetic  Poles.  —  We  are  now  in  a  position 
to  state  the  properties  which  must  be  attributed  to  magnetic 
poles  in  order  to  account  for  the  experimentally  observed  facts 
concerning  their  mutual  action.     When  the  induced  magnetic  poles 
as  well  as  the  permanent  poles  are  taken  into  account,  and  air  is 
assumed  to  be  non-magnetic  these  properties  are  the  following : 


Magnetic  Liquid 


42  ELECTRICAL  ENGINEERING 

1.  Like  poles  repel  each  other;  unlike  poles  attract  each  other. 

2.  Whenever  a  magnetic  pole  of  one  sign  exists  on  a  body  there 
is  always  an  equal  pole  of  the  opposite  sign  on  some  other  part  of 
the  same  body.     Neither  kind  of  pole  can  be  produced  by  itself. 

3.  Two  point-poles  of  "strengths"  m  and  mf  respectively  located 
a  distance  r  apart  repel  each  other  with  a  force  proportional  to  the 
product  of  the  strengths  m  and  mf  and  inversely  proportional  to  the 
square  of  the  distance  between  them;   but    independent    of    the 
medium  between  them;  that  is,  with  a  force 

ram'  (1) 

7~ 

where  fc  is  a  constant  depending  upon  the  units  in  which  m,  mf }  r, 
and  /  are  measured. 

Since  we  have  not  as  yet  specified  what  we  shall  take  as  a 
measure  of  the  strength  of  a  magnetic  pole,  we  may  select  the  unit 
of  pole  strength  so  that  this  constant  k  is  unity  when  r  is  measured 
in  centimeters  and  /  in  dynes.  Equation  (1)  then  becomes 

m  mf  .  (la) 

In  this  expression  for  the  force  between  two  magnetic  poles 
the  strengths  of  the  poles  m  and  mf  are  to  be  expressed  as  positive 
numbers  when  they  are  north  or  positive  poles  and  as  negative 
numbers  when  they  are  south  or  negative  poles.  Hence  when  m  is 
a  north  pole  and  mf  a  south  pole,  or  vice  versa,  the  force  of  repul- 
sion /  is  likewise  negative,  that  is,  the  force  /  is  an  attraction;  this 
is  in  accord  with  the  first  property  attributed  to  these  poles.  The 
expression  (la)  as  thus  understood  may  be  looked  upon  as  a 
definition  of  the  measure  of  the  strength  of  a  magnetic  pole.  For, 
in  accordance  with  this  law  of  the  mutual  action  of  two  point-poles, 
we  may  define  the  strengths  of  two  such  poles  ra  and  m'  as  equal 
when  each  repels  with  the  same  force  /,  a  third  point-pole  m" 
placed  at  the  same  distance  from  each.  A  unit  point-pole  is  then 
a  point-pole  which  repels  with  a  force  of  one  dyne  an  equal  point-pole 
placed  one  centimeter  away.  This  unit  is  called  the  c.  g.  s.  electro- 
magnetic unit  of  pole  strength. 

The  above  law  of  mutual  action  of  two  magnetic  poles  applies 
directly  only  to  point-poles,  for  when  the  poles  are  extended  over 
a  surface  the  distance  r  between  the  poles  has  no  definite  meaning, 
since  the  distance  between  every  pair  of  points  taken  respectively 


MAGNETISM        *  43 

in  the  two  surfaces  will  be  different.  However,  we  may  readily 
obtain  an  expression  for  the  mutual  action  of  two  such  poles  in  the 
following  manner.  Each  pole  may  be  considered  as  divided  into 
a  large  number  of  areas  so  small  that  they  may  be  considered  as 


Fig.   13. 

points,  and  we  may  call  the  pole  strengths  of  these  areas  mlt  m2, 
etc.,  for  the  first  pole  and  m'l}  m'2,  etc.,  for  the  second  pole,  and 
rn,  rl2,  etc.,  r21,  r22,  etc.  the  distances  between  m^  and  m'^  mv 
and  m'2,  etc.,  and  m  '2  and  ra'j,  m2  and  m'2,  etc.,  respectively. 
The  total  force  exerted  by  one  pole  on  the  other  will  then  be 


11        ,2  2  m2m 

—3—     —j-  -7-  "-*-  ~ 

'11  '12  '21  '22 

where  the  terms  of  the  right-hand  side  of  the  equation  are  added 
vectorially. 

35.  Pole  Strength  per  Unit  Area.  —  In  order  to  calculate  the 
actual  value  of  this  force  it  is  of  course  necessary  to  know  the  value 
of  the  pole  strength  of  each  of  these  elementary  surfaces;  or,  what 
amounts  to  the  same  thing,  the  pole  strength  per  unit  area  at  each 
point  of  the  surface.     Calling  ds  the  area  of  any  elementary  sur- 
face and  dm  the  pole  strength  of  this  surface,  the  pole  strength  per 
unit  area,  is  then 

dm 

"*.  (2) 

or  the  pole  strength  of  this  area  is 

dm=crds  (2a) 

There  are  experimental  methods  for  determining  the  value  of 
the  pole  strength  per  unit  area  of  a  magnetised  surface;  one 
method  is  described  in  Article  37. 

36.  Magnetic  Field  of  Force.  —  Field  Intensity  or  Magnetising 
Force.  —  A  magnetic  pole  placed  anywhere  in  the  region  of  space 
occupied  by,  or  surrounding,  a  magnet,  or  in  the  vicinity  of  an 


44  ELECTRICAL  ENGINEERING 

electric  current  (see  Chapter  III),  will  have  a  force  exerted  upon  it : 
the  region  of  space  in  which  a  magnetic  pole  is  acted  upon  by  a 
force  is  therefore  called  a  magnetic  field  of  force.  In  general,  when 
a  magnetic  pole  is  placed  in  the  vicinity  of  a  magnet  or  of  a  mag- 
netic body  this  pole  will  induce  new  poles  on  the  magnet  or  mag- 
netic body.  Again,  in  order  to  place  a  magnetic  pole  inside  a 
magnet  it  is  necessary  to  cut  a  hole  in  the  magnet,  and,  as  pointed 
out  in  Article  46,  whenever  a  hole  or  gap  is  cut  in  a  magnet, 
magnetic  poles  in  general  appear  on  the  walls  of  this  gap.  Hence, 
when  a  magnetic  pole  is  placed  near  or  inside  a  magnet  or  magnetic 
body,  new  poles  will  in  general  be  formed.  However,  the  force 
which  would  be  exerted  by  the  poles  originally  in  the  field  on  a  pole 
placed  at  any  point  in  the  field,  whether  inside  or  outside  the  mag- 
net, may  be  calculated  from  the  fundamental  law  of  the  mutual 
action  of  two  magnetic  poles  (equation  la).  Similarly,  the  force 
exerted  by  an  electric  current  on  a  magnetic  pole  may  be  calcu- 
lated from  the  fundamental  law  of  the  mutual  action  of  a  current 
and  a  pole  (see  Chapter  III).  The  force  in  dynes  which  would  be 
exerted  on  a  unit  north  point-pole  placed  at  any  point  in  a  magnetic 
field,  due  solely  to  the  poles  and  currents  originally  in  the  field,  is 
defined  as  the  intensity  of  the  magnetic  field  at  that  point  due  to  these 
agents.  The  field  intensity  at  any  point  is  also  called  the  mag- 
netising force  at  that  point. 

It  should  be  noted  that  the  unit  of  field  intensity  is  not  the 
dyne,  but  is  a  dyne  per  unit  pole;  compare  with  power,  the  unit  of 
which  is  not  the  erg,  but  is  an  erg  per  second.  No  specific  name  has 
been  given  to  the  c.  g.  s.  unit  of  field  intensity,  but  it  is  usually 
expressed  as  so  many  "  gilberts  per  centimeter,"  the  meaning  of 
which  expression  is  explained  in  Article  60 .  Magnetic  field  intensity 
or  magnetising  force  is  also  expressed  in  terms  of  "  ampere- turns 
per  centimeter  "  or  "  ampere-turns  per  inch,"  the  meaning  of 
which  expressions  is  explained  in  Chapter  IV.  The  relations  be- 
tween these  various  units  are 

1  c.  g.  s.  electromagnetic  unit        =1  gilbert  per  centimeter 
1  c.  g.  s.  electromagnetic  unit        =0.79578  ampere-turns  per 

centimeter 
1  c.  g.  s.  electromagnetic  unit        =2.0213    ampere-turns    per 

inch 

Since  the  force  in  dynes  produced  by  a  magnetic  point-pole  of 
strength  m  at  a  distance  r  centimeters  away  on  a  point-pole  of  strength 


MAGNETISM  45 

unity  is  ,  the  intensity  of  the  magnetic  field  at  a  point  P  a 

distance  of  r  centimeters  from  a  point-pole  of  strength  m,  due  solely 
to  this  pole;  is 

H=— 

whether  this  point  be  outside  or  inside  a  magnet.  The  field  in- 
tensity H  will  be  in  the  direction  of  the  line  drawn  from  m  to  P 
where  m  is  positive,  and  in  the  direction  of  the  line  drawn  from  P 
to  m  when  m  is  negative.  Field  intensity  is  therefore  a  vector 
quantity. 

The  field  intensity  at  any  point  due  to  any  number  of  point- 
poles  is  then  the  vector  sum 

H=2™  (3a) 

r2 
where  the  symbol  S  with  a  line  over  it  is  used  to  represent  a  vector 

Tfl 

sum.     Each  pole  then  contributes  a  component  —  to  the  result- 

r2 

ant  intensity,  where  m  is  the  strength  of  this  pole  and  r  is  its 
distance  from  the  given  point. 

The  force  exerted  on  any  point-pole  of  strength  m  in  a  magnetic 
field  is 

F=mH  (4) 

where  H  is  the  field  intensity  at  the  point  occupied  by  the  pole 
m  due  to  all  the  poles  in  the  field  except  the  pole  m.  The  field  due 
to  any  given  pole  can  of  course  produce  no  mechanical  force  on  this 
pole  itself;  "  a  man  cannot  lift  himself  by  his  boot-straps." 

When  the  field  intensity  has  the  same  value  at  every  point 
throughout  a  given  region,  the  field  is  said  to  be  uniform  through- 
out that  region;  we  shall  find  several  examples  of  practically  uni- 
form magnetic  fields. 

37.  Example  Illustrating  the  Application  of  Above  Definitions.— 
The  following  example  will  serve  to  illustrate  the  ideas  just  dis- 
cussed. The  problem  chosen  is  to  find  the  field  intensity 
at  any  point  on  the  axis  of  a  magnet  having  the  shape  of 
a  long  right  cylinder  and  having  its  poles  confined  entirely  to 
the  end  surfaces  of  the  cylinder  and  of  the  same  strength  per  unit 
area  at  each  point  of  these  end  surfaces.  Let  &  be  the  strength 
per  unit  area  of  the  north  pole  of  this  magnet  and  —  cr  the  pole 
strength  per  unit  area  of  its  south  pole.  Consider  first  the  field 


46 


ELECTRICAL  ENGINEERING 


intensity  clue  to  its  north  pole.  Let  the  point  P,  Fig.  14,  at  which 
the  intensity  is  to  be  determined,  be  at  a  distance  a  from  this  pole, 
and  let  the  radius  of  the  magnet  be  r.  Consider  a  ring  drawn  in  the 
end  surface  with  its  center  at  N  on  the  axis  of  the  magnet,  and 
let  the  radius  of  this  ring  be  x  and  its  width  be  dx.  The  elementary 
pole  at  any  small  area  ds  of  this  ring  will  produce  a  field  intensity 
at  P,  and  this  force  will  have  a  component  in  the  direction  NP  and 
a  component  perpendicular  to  NP.  The  elementary  pole  on  an 
equal  small  area  ds'  diametrically  opposite  ds  will  likewise  produce 
a  field  intensity  at  P  numerically  equal  to  that  produced  by  the 


Fig.    14. 

pole  at  ds,  but  in  the  direction  shown.  Hence  the  components  of 
the  field  intensities,  due  to  the  poles  at  ds  and  dsf,  perpendicular  to 
N  P  are  exactly  equal  and  opposite,  and  therefore  exactly  neutral- 
ise each  other;  similarly  for  any  other  two  diametrically  opposite 
points  in  this  ring.  Hence  the  resultant  intensity  due  to  the 
entire  ring  is  in  the  direction  N  P  and  is  equal  to 

<T  X  2  TT  xdx  a 


For  2  TT  xdx  is  the  area  of  the  ring,  and  therefore  cr  X  2  TT  zcfo  is  the 
total  pole  strength  of  the  ring;  a?  +  x*  is  the  square  of  the  distance 

of  the  point  P  from  each  element  of  the  ring,  and  — •  is  the 

cosine  of  the  angle  between  the  field  intensity  at  P  due  to  each 
point  in  the  ring  and  the  line  NP.  The  total  field  inten- 
sity Hn,  at  P,  due  to  the  entire  north  pole  of  the  magnet  is  then 
the  sum  of  the  field  intensities  due  to  all  the  contiguous  elemen- 
tary rings  into  which  the  pole  face  N  may  be  divided.  Hence 


Jx=r  r*  x  =  r 

-  =TT  a  a        (a2H 
_  («•+*•>«        J_ 


+  3?)    d(a2+x2) 


MAGNETISM  47 

=  7Tcr  a    -2(a2  +  z2P       =2  77  o-      1  -   /f 

L.  J0  L       Va2  +  r2J 

where  9  is  the  angle  between  the  axis  of  the  magnet  and  the  line 
drawn  from  P  to  the  edge  of  its  north  pole.  The  field  intensity 
due  to  the  south  pole  of  the  magnet  is  similarly 

Hs  =  -27T<T(l-COS0') 

where  0'  is  the  angle  between  the  axis  of  the  magnet  and  the  line 
drawn  from  P  to  the  edge  of  its  south  pole. 

Consequently  the  total  field  intensity  is 

H  =  Hn+  Hs=2  IT  a-  (cos  9f-cos  9}.  (5a) 

Each  end  of  this  magnet  is  a  circular  disc  over  which  is  uni- 
formly distributed  a  magnetic  pole.  It  is  interesting  to  note 
what  are  the  limiting  values  of  the  field  intensity  due  to  a  single 
magnetically  charged  disc*  when  the  point  P  is  (1)  very  close  to  the 
disc,  and  (2)  when  the  point  P  is  at  a  considerable  distance  from 
the  disc.  In  the  first  case,  cos  9  becomes  negligibly  small  in  compari- 
son with  unity,  since  9  becomes  practically  90°.  Hence  at  the 
point  on  the  axis  of  a  uniformly  magnetically  charged  disc  at  an 
infinitesimal  distance  from  its  surface,  the  field  intensity  due  to  this 
disc  alone  is 

and  is  independent  of  the  size  of  the  disc,  but  depends  only  upon 
its  pole  strength  per  unit  area. 

It  can  also  be  shown  that  a  magnetic  pole  distributed  in  any 
manner  over  any  plane  surface,  produces  a  field  intensity  at  a 
point  just  outside  this  surface  which  has  a  component  normal  to 
this  surface  at  this  point  equal  to 

Hn=27TC7  (5c) 

independent  of  the  shape  or  size  of  the  surface,  where  cr  is  the  pole 
strength  per  unit  area  of  the  surface  directly  opposite  this  point. 
In  general,  however,  there  will  also  be  a  tangential  component  to 
the  field  intensity,  that  is,  a  component  parallel  to  the  surface ;  also 
the  other  pole  of  the  magnet  will  be  sufficiently  close  to  produce  a 
field  intensity  at  this  point,  which  intensity  may  have  both  a 
normal  and  a  tangential  component. 

*e.  g.,  the  end  of  a  very  long  cylindrical  magnet  when  the  other  end  is  so 
remote  that  it  produces  no  appreciable  effect. 


48  ELECTRICAL  ENGINEERING 

When  the  point  P  is  at  a  great  distance  from  the  pole  face,  so 
that  a  is  large  compared  with  r,  the  angle  9  becomes  very  small 
and  therefore  this  angle  in  radians  is  equal  to  its  tangent,  i.e., 

a 

Whence,  expanding  cos  9  into  a  series  and  neglecting  all  terms 
of  a  higher  order  than  the  second,  we  have 

cos  9  =  1  — 

whence 

I -cos  9=— 
2a2 

and  therefore  equation  (5)  becomes 

"  cr     m 


„.  _ 


where  m=ir  r2  cr  is  the  total  pole  strength. 

Hence  the  field  intensity  at  a  point  on  the  axis  of  a  magnet  a  con- 
siderable distance  from  the  magnet  may  be  calculated  approximate- 
ly by  assuming  the  poles  concentrated  in  points  at  the  ends  of  the 
magnet.  In  most  practical  problems  a  sufficiently  accurate  value 
of  the  field  intensity  at  any  point  due  to  a  bar  magnet  may  be 
obtained  by  considering  the  poles  of  the  magnet  to  be  concentrated 
in  points  at  its  ends,  except  when  the  point  under  consideration 
is  very  close  to  a  pole,  but  the  limitations  of  this  assumption  should 
always  be  borne  in  mind. 

38.  Force  Required  to  Separate  Two  Equal  and  Opposite  Poles 
in  Contact.  —  A  useful  application  of  equation  (56)  is  the  calcu- 
lation of  the  force  required  to  separate  two  equal  and  opposite 
poles  distributed  uniformly  over  plane  surfaces,  such  as  the  two 
poles  of  two  equal  bar  magnets  placed  end  to  end.  Call  the  two 
surfaces  A  and  B  and  let  them  be  separated  by  an  infinitesimally 
narrow  gap.  Then,  if  cr  is  the  pole  strength  per  unit  area  of  the 
north  pole  on  the  wall  of  the  gap,  the  field  intensity  inside  the  gap 
produced  by  either  surface  is  H  =2  TT  cr,  and  therefore  this  sur- 
face attracts  each  unit  pole  on  the  other  surface  B  with  a  force  of 
2  TT  cr  dynes.  Hence,  calling  S  the  area  of  each  surface,  the  total 
pole  strength  of  the  surface  B  is  —  cr  S  and  consequently  the  sur- 
face A  attracts  the  surface  B  with  a  force  of 

(6) 


MAGNETISM  49 

dynes.  This  formula  is  deduced  on  the  assumption  that  the  only 
forces  acting  are  those  due  the  two  poles  which  are  separated;  if 
there  are  any  other  forces  present  (for  example,  the  forces  due  to 
the  other  ends  of  the  magnets  and  to  the  direct  action  of  the 
electric  current  in  the  coil  surrounding  the  magnets  if  the  latter  are 
electromagnets)  their  effect  also  must  be  taken  into  account.  The 
formula  is  also  based  upon  the  assumption  that  the  poles  are  con- 
fined solely  to  the  end  surfaces  and  are  uniformly  distributed  over 
these  end  surfaces,  which  is  practically  never  realized.  The 
formula  does  give  a  rough  means,  however,  for  determining  the 
average  value  of  the  pole  strength  per  unit  area,  since  both  the 
force  F  and  the  area  S  are  readily  measured. 

39.  The  Earth's  Magnetic  Field.  —  Since  a  magnet  suspended 
at  any  point  near  the  earth's  surface  tends  to  point  in  a  definite 
direction,  indicating  the  existence  of  a  force  acting  on  the  magnet, 
the  region  of  space  in  the  vicinity  of  the  earth  is  a  magnetic  field  of 
force.     For  points  at  a  considerable  distance  from  any  large  masses 
of  iron  or  wires  carrying  electric  currents,  the  intensity  of  the 
earth's  field  is  practically  uniform  over  a  large  area,  though  there 
is  a  considerable  variation  in  both  the  magnitude  and  the  direction 
of  the  earth's  field  with  longitude  and  latitude,  and  also  a  small 
variation  from  day  to  day  and  from  year  to  year.     The  earth's 
field  at  any  point  in  general  has  both  a  horizontal  and  a  vertical 
component    (downwards    in    the  northern  hemisphere,   approxi- 
mately).    The  horizontal  intensity  of  the  earth's  field  in  the  longi- 
tude of  Washington  and  at  latitude  45°  north  is  roughly  0.2  c.  g.  s. 
electromagnetic  units  (defined  above)  and  the  vertical  component 
0.55  c.  g.  s.  electromagnetic  units.     In  any  laboratory,  however, 
particularly  in  the  vicinity  of  electric  trolley  lines,  the  value  of 
the  magnetic  field  may  differ  considerably  from  this  and  varies 
continually. 

40.  Magnetic  Moment.  —  Equivalent  Length  of  a  Magnet.  — 
To  hold  a  magnet  in  a  uniform  magnetic  field  in  any  other  direction 
than  that  which  it  would  take  if  acted  upon  only  by  the  forces 
due  to  this  field,  requires  a  certain  torque  or  moment,  and  the 
maximum  value  of  this  moment  will  correspond  to  a  position  of 
the  magnet  at  right  angles  to  its  equilibrium  position  when  acted 
upon  only  by  the  forces  due  to  the  field.     The  ratio  of  the  value  of 
the  maximum  moment  which  can  be  exerted  by  a  uniform  mag- 
netic field  on  a  given  magnet  to  the  value  of  the  field  intensity,  is 


50  ELECTRICAL  ENGINEERING 

called  the  magnetic  moment  of  the  magnet.  In  the  ideal  case  of  a 
magnet  which  has  its  poles  concentrated  in  two  points  a  distance  / 
apart,  the  magnetic  moment  is  equal  to  ml  where  m  is  the  strength 
of  the  north  pole  of  the  magnet.  For,  the  total  force  acting  on 
each  pole  of  such  a  magnet  when  placed  in  a  uniform  field  of  inten- 
sity H  is  m  H,  and  when  the  magnet  js  at  right  angles  to  this  field 
the  moment  acting  on  it  is  m  HI,  whence  the  ratio  of  this  maximum 
moment  to  the  field  intensity  H  is  ml.  The  equivalent  length  of 
any  magnet  is  defined  as  the  ratio  of  its  magnetic  moment  to  the 
total  strength  of  its  north  pole. 

41.  Equality  of  the  Poles  of  a  Magnet.  —  Equilibrium  Position 
of  a  Magnet  in  a  Uniform  Magnetic  Field.  —  When  a  magnet  of  any 
shape  is  supported  in  a  uniform  magnetic  field,  for  example,  that 
due  to  the  earth,  in  such  a  manner  that  it  is  free  to  move  (e.g., 
floated  on  a  piece  of  cork) ,  the  magnet  takes  up  a  definite  position 
and  remains  at  rest.     Hence  the  two  poles  of  the  magnet  must  be 
of  equal  and  opposite  strength;  for,  calling  H  the  intensity  of  the 
field  and  m  and  m'  the  total  strengths  of  the  two  poles  respectively, 
the  force  acting  on  one  pole  of  the  magnet  is  mH  and  the  force 
acting  on  the  other  pole  is  m'H,  and  therefore  the , total  force 
on  the  magnet  is  (m  +  m')  H.     But  since  there  is  no  motion  of  the 
magnet  relative  to  the  earth,  which  produces  the  field,  this  total 
force  must  be  zero;  hence  m  =  —  mf.     Further,  since  there  is  no 
rotation  of  the  magnet,  its  position  of  equilibrium  must  be  such 
that  the  line  of  action  of  the  two  forces  mH  and  m' H  coincides. 
Hence,  when  the  magnet  is  in  the  shape  of  a  long  rod  or  needle,  a 
line  drawn  from  its  south  to  its  north  pole  will  give  the  direction 
of  the  field.     This  fact  is  usually  expressed  by  saying  that  the 
needle  points  in  the  direction  of  the  magnetic  field.     When  the 
motion  of  the  magnet  is  limited  to  a  definite  plane,  for  example, 
when  it  is  floated  on  a  piece  of  cork  or  mounted  on  a  pivot  of  the 
kind  used  in  a  magnetic  compass,  it  will  point  in  a  direction  cor- 
responding to  the  component  of  the  field  intensity  in  this  plane. 

42.  Measurement  of  the  Horizontal  Component  of  the  Intensity 
of  a  Magnetic  Field.  —  When  a  bar  magnet  or  magnetic  needle  is  sup- 
ported in  a  magnetic  field  in  such  a  manner  that  it  is  free  to  vibrate 
about  a  vertical  axis,  and  is  set  vibrating  about  this  axis,  it  will  oscil- 
late with  a  definite  period  depending  upon  the  horizontal  compo- 
nent of  the  intensity  of  the  magnetic  field ;  due  to  the  friction  of  the 
air  and  of  its  support  it  will  ultimately  come  to  rest  in  line  with 
the  horizontal  component  of  the  field  intensity.      Let  this  hori- 


MAGNETISM  51 

zontal  component  be  H  (Fig.  15), 

and  let   the   magnet   be  so   small 

that  the  intensity   H  is   constant 

over  the  space  occupied  by  it.    Let 

the  magnetic  moment  of  the  mag- 

net be  M  ,  and  the  angular  displace- 

ment of  the  magnet  from  its  posi- 

tion of  equilibrium  (the  direction  of 

H)  at  any  instant  be  0.     Let  m  be 

the  numerical  value  of  the  strength 

of  each  pole  of  the  magnet,  and  I  its  Flg-  15t 

equivalent  length.    Then  the  torque  acting  on  the  magnet,  neglect- 

ing the  friction  of  the  air  and  the  support,  tending  to  bring  it 

back   to   its   position    of   equilibrium    is   ml  H  sin  0.      When    0 

is  small,  sin  0  is  approximately  equal  to  0,  and  the  torque  is  then 

approximately    ml  HO    or    MHO.      This    torque   is    in   such   a 

direction  as  to  oppose  an  increase  in  the  angular  displacement  of 

the  magnet,  consequently  the  angular  acceleration  of  the  magnet 

about  the  vertical  axis,  neglecting  the  damping  effect  due  to 

friction,  is 


where  /  is  the  moment  of  inertia  of  the  magnet  (see  Article  19). 
This  equation  shows  that  the  motion  of  the  magnet  is  harmonic 
(see  Article  23)  i.e.,  the  magnet  will  oscillate  about  its  equilib- 
rium position  with  a  period  equal  to 


M  H 
The  frequency  with  which  the  magnet  vibrates  is  then 

~M~H 


whence  the  intensity  of  the  horizontal  component  of  the  field  is 

H     477"2//2  m 

~w~  (7) 

Note  that  the  intensity  H  is  proportional  to  the  square  of  the 
frequency  of  vibration  of  the  magnet.  Hence,  if  the  frequencies 
of  vibration  of  the  same  magnet  when  suspended  successively  in 
two  magnetic  fields  are  /i  and  /2,  the  ratio  of  the  horizontal  in- 
tensities of  these  two  fields  is 

I*lJjL 
H*     ft 

which  gives  a  simple  method  of  comparing  the  horizontal  inten- 
sities of  two  magnetic  fields. 

In  order  to  calculate  H  from  the  formula  (7),  it  is  necessary  to 
know  in  addition  to  the  frequency  of  vibration  f,  the  moment  of 


52  ELECTRICAL  ENGINEERING 

inertia  /  and  the  magnetic  moment  M  of  the  magnet.  The 
moment  of  inertia  /  can  be  readily  calculated  when  the  magnet  is  in 
the  form  of  a  bar  of  rectangular  cross  section  or  in  the  form  of  a 
right  cylinder,  or  it  may  be  determined  experimentally.  A  second 
relation  between  the  moment  M  and  the  field  intensity  H  can  be 
obtained  from  the  following  experiment.  Remove  the  bar  magnet 
from  the  point  P  and  suspend  in  its  place  a  small  magnetic  needle 
(Fig.  16).  Place  the  bar  magnet  with  its  center  at  a  distance  a 
from  P  with  its  axis  perpendicular  to  the  direction  of  H  and  in 
the  same  horizontal  plane  as  the  point  P.  The  magnetic  needle 
will  now  be  acted  upon  by  two  magnetic  fields,  the  original  field  H 
and  the  field  due  to  the  magnet.  The  intensity  of  this  latter 
field  will  be 

m  m 

fi  = 


H)' 


2  ml 


771 


Fig.   16. 

I  I  I  \2 

When  a  is  chosen  large  in  comparison  with  —  then  i^~j  will  be 

practically  negligible  in  comparison  with  unity;  hence  to  a  close 
approximation 

_2ml_2M 

~  a3   "=~tf 

where  M  is  the  moment  of  the  bar  magnet.  When  the  magnetic 
needle  is  taken  sufficiently  small  the  value  of  the  field  intensity  at 
the  needle  due  to  the  bar  magnet  will  be  practically  uniform  over 
the  space  occupied  by  the  needle,  and  therefore  the  resultant  field 
intensity  at  the  needle  will  also  be  uniform  and  will  make  an  angle 

0=tan'1  -jj-  with  the  direction  of  the  original  field  H.     This  angle 

may  be  readily  measured  by  noting  the  deflection  of  the  needle 
when  the  bar  magnet  is  removed  to  a  great  distance,  for  when 


MAGNETISM  53 

the  bar  magnet  is  present,  the  needle   points  in  the   direction 
of  this  resultant  field  intensity,  h  +  H,  and  when  the  bar  magnet 
is  removed  the  needle  points  in  the  direction  of  the  original  in- 
tensity H.     Hence,  the  value  of  h  in  terms  of  0  and  H  is 
h  =  H  tan  0 

whence  2M      u 

—  =  H  tan  0 
a3 

or  M=\a*Hian0 

which  substituted  in  the  above  equation  for  H  gives 

~2T  (7a) 


a  tan  0 

and  all  the  quantities  in  the  right-hand  member  can  be  measured 
or  calculated.  Similarly,  the  substitution  .of  this  value  for  H  in 
the  equation  just  deduced  for  M  gives 

(76) 


from  which  the  magnetic  moment  may  be  calculated. 

When  the  horizontal  component  of  the  field  intensity  has  thus 
been  determined,  the  direction  and  numerical  value  of  the  result- 
ant intensity  can  be  found  by  placing  at  the  point  P  a  magnetic 
needle  which  can  turn  only  about  a  horizontal  axis  (a  so-called 
"  dip  circle  "),  and  setting  this  horizontal  axis  perpendicular  to 
the  horizontal  component  to  the  field.  This  needle  will  then  point 
in  the  direction  of  the  resultant  field  intensity,  and  if  the  angle  it 
makes  with  the  horizontal  be  a,  the  value  of  the  resultant  inten- 
sity is 


(7C) 


cos  a 

It  should  be  noted  that  there  are  other  and  more  convenient 
methods  for  the  measurement  of  the  intensity  of  a  magnetic  field, 
based  on  the  measurement  of  an  elec- 
tric current.     But  since  as  the  measure 
of  an  electric  current  is  taken  the  force 
produced  by  a  magnetic  field  on  a  wire 
in  which  the  current  is  established,  it  is 
necessary  to  start  with  an  independent 
method  for  the  measurement  of  mag- 
netic field  intensity. 

43.  Flux  of  Magnetic  Force  Due 
to  a  Single  Pole. —  Lines  of  Magnetic 
Force  Due  to  a  Single  Pole.  —  Con- 
sider a  point-pole  m  at  any  point  P 
(Fig.  17).  The  field  intensity  at  any  Fig.  17 

point  Q  at  a  distance  r  from  m  is  —  and  is  in  the  direction  PQ 


54  ELECTRICAL  ENGINEERING 

when  the  pole  is  positive  or  in  the  direction  QP  when  the  pole 
is  negative.  Consequently  at  every  point  in  the  surface  of  a 
sphere  of  radius  r  drawn  about  m  as  a  center  the  field  intensity 

will  have  the  same  value  —  and  will  be  normal  to  the  surface  of 

r* 

the  sphere.  Hence  the  product  of  the  area  of  this  sphere  and 
the  field  intensity  at  its  surface  is 

,   m 
4  77  r.—  =4  77  m 

which  is  independent  of  the  radius  of  the  sphere  but  depends  only 
upon  the  strength  of  the  pole  m.  The  product  of  the  area  of  any 
sphere  drawn  about  a  point-pole  as  its  center  and  the  field  intensity 
at  the  surface  of  this  sphere  is  called  the  flux  of  force  due  to  this 
pole,  and  this  flux  of  force  is  said  to  be  outward  from  the  pole  when 
the  pole  is  positive  and  inward  towards  the  pole  when  the  pole  is 
negative.  A  magnetic  pole  of  strength  m  then  produces  a  flux  of 

f°rCe  l//=477W  (8) 

outward.  (When  m  is  negative  the  flux  outward  is  also  negative, 
which  is  equivalent  to  a  positive  flux  inward.) 

The  flux  of  force  from  a  magnetic  point-pole  can  be  represented 
graphically  in  a  simple  manner.  Imagine  the  surface  of  any 
sphere  surrounding  the  pole  divided  into  4  TT  m  equal  areas  and 
cones  drawn  with  these  areas  as  their  bases  and  their  common 
vertex  at  the  pole  m,  and  let  the  lateral  walls  of  these  cones  extend 
out  indefinitely.  The  number  of  these  cones  then  represents  the 
flux  of  force  from  the  pole  m.  Each  of  these  cones  may  be  repre- 
sented by  a  line  coinciding  with  its  axis,  and  the  number  of  these 

lines  will  also  represent  the  flux  of 
force  from  the  pole  m,,  and  these 
lines  will  coincide  in  direction  with 
the  field  intensity  at  every  point  in 
their  path.  Such  lines  are  called 
magnetic  lines  of  force. 

From  the  definition  of  these 
lines  of  force,  it  follows  that  the 
number  of  these  lines  per  square 
centimeter  of  area  normal  to  their 
direction  at  any  point  will  be  equal 
to  the  field  intensity  at  that  point; 


MAGNETISM  55 

for  the  unit  area  normal  to  their  direction  at  any  point  Q  at  a  distance 
r  from  the  pole  will  be  unit  area  in  the  surface  of  a  sphere  of  radius 
r,*and  since  a  total  of  4  TT  m  equally  spaced  lines  are  considered  as 
radiating  out  from  the  pole  m,  the  number  crossing  this  unit  area 
is  equal  to  the  total  number  of  these  lines  divided  by  the  total 

surface  of  the  sphere,  that  is =— .     Hence  the  lines  of  force 

471-r2     r2 

due  to  a  point-pole  m  coincide  in  direction  at  every  point  with 
the  direction  of  the  field  intensity  at  that  point  due  to  this  pole 
and  the  number  of  these  lines  crossing  unit  area  at  right  angles  to 
their  direction  is  equal  to  the  field  intensity  at  this  area  due  to 
the  pole  m.  The  lines  due  to  a  single  north  pole  radiate  out  to 
infinity;  those  due  to  a  south  pole  radiate  in  from  infinity. 

It  should  be  borne  in  mind  that  these  lines  really  represent 
cones,  and  therefore  it  is  entirely  logical  to  speak  of  a  fraction  of 

a  line  of  force.     For  example,  from  a  pole  of  strength  - —  there 

4?r 

would  -be  but  a  single  cone,  which  would  fill  all  space;  that  is, 
the  cone  would  be  a  sphere  with  m  at  its  center.  A  unit  area 
normal  to  the  direction  of  the  field  intensity  at  a  distance  of  10 

centimeters    from    this    pole   would    cut    out = -th 

47TX102      1256 

of  this  cone,  or  this  unit  area  would  be  cut  by  only   -      — th  of 

this  cone  of  force,  which,  from  the  manner  in  which  these  cones 
are  drawn,  is  equal  to  the  field  intensity  at  this  point. 

A  simple  relation  also  exists  between  the  number  of  these  lines 
of  force  crossing'  any  area  and  the  field  intensity  normal  to  that 
area.  Let  ds  (  Fig.  19)  be  any  plane  area  at  any  point  P  taken 


Fig.   19. 


so  small  that  the  field  intensity  at  every  point  in  this  area  may  be 
considered  as  having  the  same  value  and  the  same  direction,  and 


56  ELECTRICAL  ENGINEERING 

let  the  normal  to  this  area  make  the  angle  a  with  the  direction  of 
the  field  intensity,  that  is,  with  the  direction  of  the  lines  of 
force  through  ds.  Let  ds'  be  the  projection  of  ds  on  a  plane 
through  P  perpendicular  to  the  direction  of  the  field  intensity. 
Then  the  number,  d\jj,  of  lines  of  force  crossing  ds'  must  be  the  same 
as  the  number  crossing  ds,  since  the  perpendiculars  dropped  from 
the  periphery  of  ds  on  the  plane  of  ds'  are  parallel  to  the 
lines  of  force.  But  the  field  intensity  H  at  the  point  P  is  equal 
to  the  number  of  lines  of  force  per  unit  area  crossing  ds',  that  is 

^~ds' 
But  ds'  =ds.  cos  a,  hence 

u          d*b 

£1  =• — 


ds.  cos  a 

u  d*l> 

or  H  cos  a=—!- 

ds 

Therefore  the  number  of  lines  of  force  per  unit  area  at  any  point  in 
a  magnetic  field,  due.  to  a  point-pole  of  strength  m,  is  equal  to  the 
component  of  the  field  intensity  at  this  point  normal  to  this  area. 

An  important  point  to  be  noted  in  regard  to  a  line  of  force  is 
that  a  line  crossing  a  surface  in  the  direction  from  the  side  A  to  the 
side  B  is  to  be  considered  as  equivalent  to  a  negative  line  crossing 


180-oc 


Fig.   20. 

this  surface  in  the  direction  from  the  side  B  to  the  side  A.  For, 
calling  a  the  angle  between  the  direction  of  the  field  intensity  H 
and  the  normal  drawn  outward  from  B,  then  the  angle  between 
the  direction  of  the  field  intensity  and  the  normal  drawn  outward 
from  A  is  180°— a.  Hence  the  number  of  lines  of  force  crossing 
the  surface  ds  is  (H  cos  a)  ds  in  the  direction  from  A  to  B,  or  is 
[  H  cos  (180°  —  a)]  ds  =  —  ( H  cos  a)  ds  in  the  direction  from  B  to  A . 
44.  Gauss's  Theorem.  —  An  extremely  useful  relation  in  the 


MAGNETISM  57 

theory  of  magnetism  is  that  the  algebraic  sum  of  the  lines  of  force 
outward  across  any  closed  surface  of  any  shape  whatever  is  equal 
to  4  77  times  the  algebraic  sum  of  the  strengths  of  all  the  poles 
inside  this  surface.  This  follows  immediately  from  the  fact 
that  from  any  north  pole  of  strength  m  there  radiate  out  4  77  m 
lines  of  force  and  therefore  all  these  4  IT  m  lines  of  force  from 
a  north  pole  inside  such  a  surface  will  cut  this  surface  in  the  direc- 
tion from  the  inside  outward,  while  to  a  south  pole  of  strength  m' 
there  radiate  in  4  77  m'  lines  of  force,  and  therefore  all  these  4  77  m' 
lines  of  force  radiating  into  a  south  pole  inside  such  a  surface  cut 
this  surface  in  the  direction  from  the  outside  inward.  Also,  the 
lines  of  force  due  to  any  pole  outside  this  surface,  whether  the  pole 
be  a  north  or  a  south  pole,  will  either  not  cut  the  surface  at  all  or 
will  cut  it  an  even  number  of  times,  as  many  times  in  the  direction 
from  outside  inward  as  in  the  direction  from  inside  outward. 
Since  a  line  of  force  inward  is  equivalent  to  a  negative  line  of  force 
outward,  in  calculating  the  algebraic  sum  of  the  lines  of  force  out- 
ward the  lines  of  force  due  to  the  south  poles  inside  the  surface  are 
to  be  substracted  from  the  lines  of  force  due  to  the  north  poles  in- 
side the  surface ;  the  lines  of  force  due  to  any  pole  outside  the  sur- 
face contribute  nothing  to  the  algebraic  sum  of  the  lines  of  force 
outward,  since  each  line  leaves  the  surface  as  many  times  as  it 
enters  it.  Hence  the  algebraic  sum  of  the  lines  of  force  which  cut 
a  closed  surface  in  the  direction  from  the  inside  outward  is  equal 
to  4  TT  times  the  algebraic  sum  of  the  strengths  of  all  the  poles  inside 
this  surface. 

45.  Lines  of  Force  Representing  the  Resultant  Field  Due  to 
any  Number  of  Magnets.  —  So  far  we  have  been  considering  the 
lines  of  force  due  to  each  point-pole  as  a  separate  set  of  lines,  there 
being  as  many  sets  of  these  lines  as 
there  are  individual  point-poles.  In 
any  actual  case,  however,  whenever 
there  is  one  magnetic  pole  there 
must  be  somewhere  else  in  the  field 
an  opposite  pole  of  equal  strength. 
That  is,  in  any  actual  case,  the  sum 
of  the  strengths  of  all  the  north 
poles  in  the  field  is  equal  to  the  sum 
of  the  strengths  of  all  the  south  ds 
poles  in  the  field.  A  field  due  to  Fig.  21. 


58  ELECTRICAL  ENGINEERING 

any  number  of  such  equal  and  opposite  poles  may  also  be  repre- 
sented by  a  single  set  of  lines,  drawn  out  from  the  north  poles  in 
the  field,  4  TT  of  these  ''resultant7'  lines  being  drawn  from  each  unit 
north  pole,  and  each  line  coinciding  in  direction  at  each  point 
with  the  resultant  field  intensity  at  that  point.  All  of  these  lines 
will  also  end  on  south  poles,  4  TT  of  them  on  every  unit  south  pole. 
Also,  the  number  of  these  resultant  lines  of  force  per  unit  area 
crossing  any  elementary  area  ds  at  any  point  in  the  field  is  equal 
to  the  component  Hn  of  the  resultant  field  intensity  normal  to 
this  area,  i.e. 

HnJ^  <9> 

ds 

when  the  area  ds  is  taken  normal  to  the  direction  of  the  field  in- 
tensity the  number  of  lines  of  force  per  unit  area  is  equal  to  the 
resultant  intensity  H,  since  in  this  case  the  normal  component  is 
equal  to  the  resultant. 

Consider  first  the  number  of  lines  of  force,  considered  as  sepa- 
rate sets  of  radial  lines  from  each  pole,  which  cross  any  elemen- 
tary area  ds  in  a  magnetic  field  due  to  any  number  of  poles.  In 
Fig.  21  let  H1  H2  etc.,  be  the  field  intensities  at  ds  due  to  the  in- 
dividual poles  producing  the  field,  and  let  a,,  a2,  etc.,  be  the  angles 
between  the  normal  to  ds  and  the  directions  of  the  respective  field  in- 
tensities at  ds.  Then  the  number  of  lines  of  force  crossing  ds  in  the 
direction  of  this  normal  to  ds  due  to  the  first  pole  is  (H1  cos  aj 
ds;  the  number  of  lines  of  force  crossing  ds  in  this  same  direction 
due  to  the  second  pole  is  (H2  cos  a2)  ds;  etc.  Therefore  the  total 
number  of  these  lines  of  force  crossing  ds  in  direction  of  the  normal 
is 

d\fy  —  (Hl  cos  a>i  +  H2cos  a2  +  -        —  )  ds 
But  Hl  cos  at  -f  H  2  cos  a2  +—  —  =H  cos  a 

where  H  is  the  resultant  field  intensity  at  ds  and  a  is  the  angle 
between  the  normal  to  ds  and  the  direction  of  this  resultant  field 
intensity.  Hence  the  net  number  of  lines  of  force  crossing  ds  in 
this  direction  may  also  be  written 

d\jj  =  (Hcosa)ds  (10) 

and  therefore  the  net  number  of  lines  of  force  crossing  any 
surface  S  is 

(Hcosa)ds  (10a) 


/» 

=    I 
J 


m 
where  H  is  the  resultant  field  intensity  at  any  element  ds  of  this 


MAGNETISM 


59 


surface  and  a  is  the  angle  between  the  outward  normal  to  this  sur- 
face at  ds  and  the  direction  of  the  resultant  field  intensity  at  ds, 


1  I     indi 
J  s 


and  I     indicates  the. sum  or  integral  of  (H  cos  a)ds  for   all  the 

J  s 
elementary  areas  ds  into  which  the  surface  is  divided. 

To  prove  that  the  resultant  field  intensity  H  may  be  repre- 
sented by  a  single  set  of  lines,  let  a  closed  surface  A  (Fig.  22)  be 
drawn  in  the  field  in  such  a  manner  that  it  encloses  all  the  north 
poles  but  none  of  the  south  poles.  Then,  by  Gauss's  Theorem,  the 


Fig.  22. 


algebraic  sum  of  the  lines  of  force  coming  out  from  this  surface  ,is 
4  TT  times  the  total  strength  of  all  the  north  poles  inside  this  sur- 
face. Hence,  calling  M  the  total  strength  of  all  the  north  poles  in 
the  field,  we  have  from  equation  (10a)  that 


J. 


(H  cos  a)  ds=47rM. 

A 

Divide  this  surface  A  into  4  TT  M  areas  such  that  the  integral  of 
( H  cos  a)  ds  over  each  of  these  areas  is  equal  to  unity,  that  is,  such 
that  the  net  number  of  lines  of  force  coming  out  through  each  of 
these  areas  is  unity.  Through  the  perimeters  of  these  areas  draw 
tubes  in  such  a  manner  that  the  lateral  walls  of  each  tube  are 
tangent  at  each  point  to  the  direction  of  the  resultant  field  intensity 
at  that  point.  These  tubes  will  fill  all  space.  No  tube  can  cross 
another  since  the  resultant  field  intensity  can  have  but  a  single 
direction  at  any  one  point,  and  by  hypothesis  the  walls  of  these 
tubes  are  tangent  at  every  point  to  the  resultant  field  intensity  at 


60  ELECTRICAL  ENGINEERING 

that  point.  Let  a  second  surface  B  be  drawn  in  the  field  in  such 
a  manner  that  it  encloses  all  the  south  poles  but  none  of  the  north 
poles.  Let  S  be  the  area  cut  out  of  the  surface  A  by  any  one  of 
these  tubes,  and  S'  the  area  cut  out  of  B  by  this  tube. 

Consider  the  closed  surface  formed  by  the  lateral  walls  of  this 
tube  and  the  two  areas  S  and  S'.  Since  there  are  no  poles  between 
S  and  S',  we  have  from  Gauss's  Theorem  that  the  net  number  of 
lines  of  force  leaving  this  closed  section  of  the  tube  is  zero.  But 
the  net  number  of  lines  of  force  crossing  the  lateral  walls  of  this  tube 
is  zero,  since  for  each  area  ds  in  these  lateral  walls  ( H  cos  a)  ds  — 
(  H  cos  90°)  ds  =0.  But  this  tube  is  drawn  in  such  a  manner  that 
entering  it  at  S  the  net  number  of  lines  of  force  is  unity,  since  the 
integral  of  ( H  cos  a)  ds  over  this  surface  S  is  unity,  where  a  is  the 
angle  between  the  direction  of  the  resultant  field  intensity  at  ds 
and  the  normal  drawn  from  ds  into  the  tube.  Hence  the  net 
number  of  lines  of  force  leaving  the  tube  at  S'  must  also  be  unity. 
Consequently  the  number  of  these  tubes  entering  the  closed  sur- 
face B  surrounding  all  the  south  poles  is  equal  to  the  net  number 
of  lines  of  force  (considered  as  individual  sets  of  lines)  entering 
this  closed  surface.  But  since  the  total  strength  of  all  the  south 
poles  inside  this  surface  B  is  equal  to  the  total  strength  of  all  the 
north  poles  inside  the  surface  A,  the  total  number  of  these  tubes 
entering  the  closed  surface  B  is  equal  to  4  TT  M.  Hence  all  the 
tubes  leaving  the  closed  surface  A  surrounding  the  north  poles 
enter  the  closed  surface  B  surrounding  the  south  poles.  This 
same  relation  holds  when  the  two  surfaces  A  and  B  are  drawn 
infinitely  close  to  the  north  poles  and  south  poles  respectively; 
consequently  4  TT  of  these  tubes  must  originate  at  every  unit 
north  pole  in  the  field  and  4  TT  of  them  end  on  every  unit  south 
pole  in  the  field.  Also,  the  net  number  of  lines  of  force  cross- 
ing any  surface  in  the  field  other  than  that  on  which  there  is  a 
pole,  is  equal  to  the  number  of  these  tubes  crossing  this  surface. 

Hence  these  tubes  are  mathematically  identical  with  the  lines 
of  force  considered  as  separate  sets  radiating  from  or  to  each 
individual  point-pole  in  the  field,  and  therefore  if  we  represent  each 
of  these  tubes  by  a  line  coinciding  with  its  axis,  we  may  consider 
these  latter  lines  as  "  resultant  "  lines  of  force.  From  equation 
(10)  the  number  d\l/  of  these  resultant  lines  of  force  per  unit  area 
crossing  any  elementary  area  ds  at  any  point  in  the  field  is  equal 
to  the  component  of  the  resultant  field  intensity  normal  to  this 
area,  that  is 

HcosaJ^  (106) 

ds 

where  H  is  the  resultant  field  intensity  and  a  is  the  angle  between 
the  normal  to  this  area  and  the  direction  of  the  resultant  field 
intensity  at  this  point.  When  the  elementary  area  is  taken  normal 
to  the  direction  of  these  lines  the  resultant  field  intensity  is  equal 


MAGNETISM  61 

to  the  number  of  these  lines  per  unit  area  crossing  this  elementary 
area,  that  is 


where  dsn  represents  an  elementary  area  normal  to  the  direction 
of  these  resultant  lines  of  force. 

It  should  be  noted  that  the  idea  of  lines  of  force  is  based  upon 
the  idea  of  field  intensity,  and  consequently  the  statement  that 
the  number  of  resultant  lines  of  force  per  unit  area  perpendicular 
to  their  direction  is  equal  to  the  field  intensity  is  not  a  definition. 
A  definition  presupposes  a  knowledge  of  all  the  terms  in  it,  and 
consequently  the  use  of  this  statement  as  a  definition  is  a  species 
of  "  arguing  in  a  circle."  The  definition  of  field  intensity  at  any 
point  is  the  force  in  dynes  due  to  the  agents  producing  the  field 
that  would  be  exerted  upon  a  unit  north  point-pole  placed  at  that 
point  (see  Article  36).  These  resultant  lines  of  force  are  simply 
a  convenient  means  of  representing  the  resultant  magnetic  field, 
these  lines  being  drawn  in  such  a  manner  that  their  direction  at 
each  point  coincides  with  the  direction  of  the  resultant  field  in- 
tensity at  that  point  and  their  number  per  unit  area  at  that  point 
crossing  a  surface  perpendicular  to  their  direction  is  equal  to  the 
resultant  field  intensity  at  that  point. 

A  rough  picture  of  the  direction  of  the  lines  of  force  represent- 
ing the  horizontal  component  of  the  resultant  field  outside  any 


Fig.  23. 


number  of  magnets  can  readily  be  obtained  by  placing  the  magnets 
on  a  table  and  sprinkling  fine  iron  filings  over  the  table  in  their 


62  ELECTRICAL  ENGINEERING 

vicinity.  When  the  table  is  tapped  lightly,  the  filings  arrange 
themselves  in  the  direction  of  the  horizontal  component  of  the 
field  intensity.  This  is  due  to  the  fact  that  the  filings  are  mag- 
netised by  induction  and  each  then  acts  like  a  small  magnetic 
needle,  setting  itself  in  the  direction  of  the  component  of  the  field 
intensity  parallel  to  the  surface  of  the  table.  It  should  be  noted, 
however,  that  in  general  the  field  intensity  has  a  value  inside  the 
magnets  also  and  that  therefore  there  are  lines  of  force  inside  the 
magnets  as  well  as  outside. 

In  Fig.  23  are  shown  the  lines  of  force  due  to  a  north  point- 
pole  and  an  equal  south  point-pole  considered  as  two  separate  sets 
of  lines,  one  set  radiating  out  from  the  north  pole  and  one  set 
radiating  into  the  south  pole,  and  in  Fig.  24  are  shown  the  result- 
ant lines  of  force  due  to  these  same  two  poles.  In  each  case  the 
lines  are  drawn  so  that  their  number  per  unit  area  at  any  point 
perpendicular  to  their  direction  is  proportional  to  the  field  intensity 
at  this  point. 

From  the  definition  of  the  lines  of  force  representing  the  re- 
sultant field,  it  follows  that  where 
the  field  intensity  is  great  these  lines 
of  force  will  be  close  together,  and 
where  the  field  intensity  is  weak 
these  lines  of  force  will  be  far  apart. 
Again,  since  4?rof  the  lines  originate 
from  each  unit  north  pole  and  end 
on  each  unit  south  pole,  it  follows 
that  these  lines  will  be  close  together 
Fig-  24-  where  they  enter  or  leave  a  surface 

on  which  the  pole  strength  per  unit  area  is  large,  and  will  be  far 
apart  where  they  enter  or  leave  a  surface  on  which  the  pole  strength 
per  unit  area  is  small.  Again,  since  a  surface  has  two  sides,  these 
lines  will  in  general  leave  both  sides  of  a  surface  on  which  there 
is  a  north  pole  and  enter  both  sides  of  a  surface  on  which  there 
is  a  south  pole. 

46.  Intensity  of  Magnetisation.  —  When  a  narrow  gap  is  cut  in 
a  magnet,  either  permanent  or  induced,  it  is  found  that  in  general 
poles  appear  upon  the  walls  of  this  gap  and  that  these  poles  are 
equal  and  opposite,  a  north  pole  appearing  on  the  wall  of  the  gap 
nearer  the  south  pole  of  the  original  magnet,  and  a  south  pole  on 
the  wall  of  the  gap  nearer  the  north  pole  of  the  original  magnet. 


MAGNETISM  63 

The  strength  per  unit  area  of  the  poles  which  appear  on  the  wall 
of  such  a  gap  is  found  to  depend  upon  the  direction  in  which  the 
gap  is  cut.  There  is  one  direction  at  each  point  in  the  magnet 
for  which  this  pole  strength  per  unit  area  is  a  maximum,  and 
when  the  gap  is  cut  perpendicular  to  this  direction  no  poles  appear 
on  its  walls.  (In  the  case  of  a  long  bar  magnet,  a  gap  cut  ^at 
its  center  perpendicular  to  its  axis  will  have  maximum  pole 
strength  per  unit  area,  while  on  a  gap  cut  at  this  point  parallel 
to  its  axis  there  will  be  no  poles  formed.) 

A  magnetised  body  may  then  be  considered  as  made  up  of 
magnetic  filaments  such  that  were  the  lateral  walls  of  any  one  of 
these  filaments  separated  from  the  rest  of  the  magnet  by  a  narrow 
air  gap,  no  poles  would  be  formed  on  these  lateral  walls.  The 
two  ends  of  any  such  filament  where  it  terminates  in  the  surface 
of  the  magnetised  body  must  then  have  equal  and  opposite  mag- 
netic poles ;  if  the  filament  is  cut  transversely  by  a  narrow  air  gap 
at  any  point,  equal  and  opposite  poles  will  be  formed  at  the  two 
walls  of  this  air  gap,  and  these  poles  will  in  turn  be  numerically 
equal  to  the  poles  at  the  ends  of  the  filament  in  the  original  sur- 
face of  the  magnetised  body.  (For,  since  each  filament  is  a  mag- 
net, it  must  have  equal  and  opposite  poles,  and  by  definition  each 
filament  has  no  poles  on  its  lateral  walls.)  Such  a  filament  is  con- 
sidered as  existing  only  in  a  magnetised  body;  the  filament  is 
broken  by  the  transverse  air  gap  just  as  a  string  is  broken  when  it 
is  cut  in  two. 

Consider  such  a  filament  cut  at  any  point  by  an  air  gap  per- 
pendicular to  its  axis,  and  let  dsn  be  its  cross  section  at  this  point, 
and  let  dm  be  the  pole  strength  of  the  north  pole  formed  where  it 
ends  in  the  gap.  Then  the  pole  strength  per  unit  area  of  this  pole 
is 

dm 
dsn 

or  vice  versa,  the  strength  of    -am 
the  pole  dm  formed  where  the 
filament  is  cut  by  the  gap  is 

dm=(rndsn 
If  the  filament  is    broken    at 

any  other  point,  poles  of  exactly  the  same  strength  will  be  formed 
on  the  broken  ends,  a  north  pole  on  the  end  nearer  to  the  original 
south  pole  of  the  magnetised  body  and  a  south  pole  on  the 


64  ELECTRICAL  ENGINEERING 

opposite  end.  Also,  the  strength  of  the  poles  where  this  filament 
ends  in  the  original  surface  of  the  magnetised  body  will  be 
numerically  equal  to  dm.  Consequently  where  the  cross  section 
dsn  of  the  filament  is  great  the  strength  per  unit  area  of  the  pole 
which  would  be  formed  were  it  broken  in  two  is  small,  and  where 
the  cross  section  is  small  the  pole  strength  per  unit  area  is  large. 
The  strength  per  unit  area  of  the  pole  which  would  be  formed  on 
the  walls  of  a  gap  cut  at  any  point  in  a  magnetised  body  perpen- 
dicular to  the  direction  of  the  magnetic  filaments  of  which  the 
body  may  be  considered  as  made  up  is  defined  as  the  intensity  of 
magnetisation*  of  the  body  at  this  point,  and  may  be  represented 
by  the  symbol  J.  That  is, 

,     dm 

J=       =an  (11) 

dsn 

where  dsn  is  the  cross  section  of  the  filament  at  the  point  under 
consideration  and  dm  is  the  strength  of  the  north  pole  which  would 
be  formed  on  one  wall  of  a  narrow  gap  coinciding  with  the  cross 
section  dsn.  The  direction  of  the  intensity  of  magnetisation  J  is 
chosen  arbitrarily  as  the  direction  of  the  magnetic  filament  at 
this  point  and  the  positive  sense  of  J  is  chosen  as  the  sense  of  the 
line  drawn  into  the  gap  from  the  wall  of  the  gap  on  which  the  north 
pole  is  formed. 

When  a  magnetic  filament  is  cut  by  a  gap  ds  which  is  not  at 
right  angles  to  the  axis  of  the  filament,  the  numerical  value  of  the 
strength  of  the  pole  formed  on  either  wall  of  the  gap  must  be  equal 
to  the  strength  of  the  pole  which  would  be  formed  on  a  gap  dsn 
cut  normal  to  the  filament,  since  the  pole  formed  on  either  gap 
must  be  equal  to  the  strength  of  the  pole  at  either,  end. of  the  fila- 
ment in  the  original  surface  of  the  magnetised  body.  Let  <r  be  the 
pole  strength  per  unit  area  of  the  north-pole  end  of  the  filament 
in  the  surface  of  the  gap,  let  a  be  the  angle  between  the  direction 
of  this  filament  and  the  normal  drawn  outward  into  the  gap  from 

*Intensity  of  magnetisation  at  any  point  may  also  be  defined  as  the  mag- 
netic moment  per  unit  volume  of  an  infinitesimal  length  of  the  magnetic 
filament  passing  through  that  point.  For,  calling  dsn  the  cross  section  of 
the  filament,  (Tn  the  numerical  value  of  the  pole  strength  per  unit  area  at 
each  end  of  the  infinitesimal  length  dl  of  the  filament;  the  magnetic  moment 
of  this  element  of  the  filament  is  (CTn  dsn)  dl  and  the  volume  is  dsndl,  whence 
the  magnetic  moment  per  unit  volume  is  (Tn ;  and  therefore  this  definition  is 
equivalent  to  that  given  above. 


MAGNETISM  65 

this  north  pole,  and  let  dsn  be  the  projection  of  ds  on  a  plane 
normal  to  the  direction  of  the  filament.  Then  from  (11) 

dm=crds=Jdsn 
But 

dsn  =ds  cos  a 
whence 

cr=Jcosa  (Ha) 

That  is,  the  pole  strength  per  unit  area  which  would  be  formed  on 
the  wall  of  a  narrow  gap  cut  in  any  direction  at  any  point  in  a 
magnet  is  equal  to  the  component  of  the  intensity  of  magnetisa- 
tion in  the  direction  of  the  normal  drawn  outward  from  this  wall 
into  the  gap. 

Intensity  of  magnetisation  may  be  represented  by  lines  just 
as  magnetic  field  intensity  may  be  represented  by  lines.  This  is 
done  by  choosing  arbitrarily  the  size  of  a  unit  magnetic  filament 
and  representing  each  unit  filament  by  a  line  coinciding  with  its 
axis.  As  the  unit  magnetic  filament  is  taken  a  filament  such  that 
were  it  cut  by  a  narrow  air  gap  at  any  point,  the  strength  of  the 

pole  formed  on  either  wall  of  the  gap  would  be  equal  to  —  .      The 

47T 

line  representing  such  a  filament  is  called  a  line  of  magnetisation; 
the  direction  of  this  line  coincides  with  the  direction  of  the  inten- 


N 


Fig.   26.     Lines  of  Magnetisation  in  a  Bar  Magnet. 

sity  of  magnetisation.  Hence  from  every  unit  south  pole  in  the 
surface  of  a  magnetised  body  4  IT  lines  of  magnetisation  originate, 
these  lines  run  through  the  magnetised  body  to  the  surface  over 
which  the  north  pole  is  distributed,  477  of  them  ending  in  every 
unit  north  pole  in  the  surface  of  the  magnet. 

The  reason  for  introducing  the  factor  —  is  to  have  number  of 

4?r 

these  filaments  leaving  a  south  pole  equal  to  the  lines  of  force 
entering  that  pole  and  the  number  of  these  filaments  entering  a  north 
pole  equal  to  the  lines  of  force  leaving  that  pole.  It  should  be 


66  ELECTRICAL   ENGINEERING 

noted  that  these  filaments  are  confined  entirely  to  magnets  or 
magnetised  bodies,  while  lines  of  force  in  general  exist  in  all  the 
space  surrounding  a  magnetic  pole,  whether  this  space  be  occupied 
by  a  non-magnetic  or  by  a  magnetic  body. 

The  relation  between  the  number  of  lines  of  magnetisation  d  N 
crossing  any  elementary  surface  ds  and  the  intensity  of  magnetisa- 
tion at  ds  is  given  by  the  formula 

dN=4:7r  (J  cos  a)  ds  (12) 

where  a  is  the  angle  between  the  direction  of  these  lines  at  ds  and 
the  direction  of  the  normal  to  ds.  J  cos  a  is  the  component  of  the 
intensity  of  magnetisation  normal  to  the  surface  ds.  Compare 
with  the  mathematical  expression  (10)  for  the  number  of  lines  of 
force  crossing  an  area. 

Since  in  the  case  of  a  long  bar  magnet  the  poles  are  confined 
almost  entirely  to  its  ends,  the  lines  of  magnetisation  inside  the 
magnet  near  its  center  must  be  parallel  to  the  sides  of  the  magnet. 
When  such  a  magnet  is  cut  in  two  by  a  plane  surface  perpendicular 
to  its  axis,  the  magnetic  poles  which  appear  on  the  walls  of  the 
gap  thus  formed  must  then,  from  equation  (11),  have  a  strength  per 
unit  area  equal  to  the  intensity  of  magnetisation  at  this  surface. 
(This  is  strictly  true  only  in  case  the  gap  between  the  two  parts  of 
the  magnet  is  of  infinitesimal  width.) 

This  fact  suggests  a  method  for  the  experimental  determination 
of  the  intensity  of  magnetisation  of  a  bar  which  can  be  separated  in 
two  parts.  For,  by  equation  (6),  the  force  required  to  separate  the 
two  equal  and  opposite  poles  which  appear  on  the  walls  of  the  gap 
formed  by  separating  the  two  parts  of  the  bar  is 


where  S  is  the  cross  section  of  the  bar  and  or  the  strength  of  these 
poles  per  unit  area,  which  may  be  assumed  constant  over  each  wall 
of  the  gap.  Hence  in  this  case  the  intensity  of  magnetisation  is 


2TTS 

and  both  F  and  S  are  readily  measured.  In  employing  this 
method  certain  corrections  have  to  be  made  for  the  effect  of  the 
action  of  other  forces  on  the  poles  which  are  separated.  A 
description  of  the  method  and  apparatus  will  be  found  in  Foster's 
Pocket  Book,  p.  94. 

(In  the  above  discussion  the  matter  forming  a  magnet  is  con- 


MAGNETISM  67 

sidered  as  absolutely  continuous,  and  no  hypothesis  is  made  as  to 
its  molecular  structure,  just  as  in  the  ordinary  theory  of  the  elastic 
properties  of  matter  a  beam  or  column  is  considered  as  made  up  of 
continuous  fibers  and  no  account  is  taken  of  the  molecular  struc- 
ture of  these  fibers. 

The  modern  theory  of  the  molecular  structure  of  a  magnet 
assumes  that  each  molecule  of  the  magnet  contains  one  or  more 
electrically  charged  particles,  which  by  their  motion  give  rise  to 
lines  of  magnetic  force  which  form  closed  loops  threading  the 
molecules  of  the  magnet.  Inside  the  magnet  these  lines  of  force 
are  in  the  direction  from  the  south  pole  of  the  magnet  to  its  north 
pole  and  outside  in  the  direction  from  its  north  pole  to  its  south ; 
the  pole  of  a  magnet  is  then  simply  an  apparent  property  possessed 
by  the  surface  of  the  magnet  where  it  is  cut  by  these  lines.  These 
closed  lines  of  force  are  then  identical  with  what  will  be  defined 
presently  as  lines  of  induction.  This  hypothesis,  while  probably 
correct,  can  be  fully  understood  only  after  one  has  become  thor- 
oughly acquainted  not  only  with  the  facts  concerning  magnetism, 
but  also  with  the  phenomena  of  electric  currents  and  electrostatics. 
We  shall  therefore  still  continue  to  consider  a  magnetic  pole  as  an 
actual  entity  which  has  the  properties  which  have  been  assigned 
to  it,  and  to  avoid  confusion,  shall  distinguish  carefully  between 
lines  of  force,  lines  of  magnetisation  and  lines  of  induction.) 

47.  Lines  of  Magnetic  Induction. —  Flux  of  Induction.  —  We 
have  seen  that  a  magnet  may  be  considered  as  made  up  of  a 
number  of  magnetic  filaments,  or  lines  of  magnetisation,  at  the 
ends  of  which  are  located  the  poles  of  the  magnet,  and  that  these 
poles  in  turn  give  rise  to  lines  of  force  equal  in  number  to  the 
number  of  lines  of  magnetisation  in  the  magnet.  The  lines  of 
magnetisation  are  confined  entirely  to  the  substance  of  the  magnet 
and  are  considered  as  originating  at  its  south  pole  and  running 
through  the  magnet  to  its  north  pole.  Lines  of  force  exist  both  in 
the  magnet  and  in  the  surrounding  space  and  are  considered  as 
originating  at  the  north  pole  of  the  magnet  and  running  through 
both  the  medium  surrounding  the  magnet  and  the  magnet  itself  to 
its  south  pole.  In  the  substance  of  the  magnet  there  are  therefore 
both  lines  of  magnetisation  and  lines  of  force,  but  the  lines  of  force 
in  a  magnet  due  solely  to  the  poles  of  this  magnet  are  in  the  opposite 
direction  to  the  lines  of  magnetisation.  In  the  case  of  an  induced 
magnet,  e.g.,  a  piece  of  soft  iron  in  a  magnetic  field  produced  by 
some  other  agent,  the  resultant  lines  of  force  are  in  general  in  the 
same  direction  as  the  lines  of  magnetisation.  In  any  case  the 
algebraic  sum  of  the  lines  of  magnetisation  and  the  lines  of  force 
crossing  any  surface  in  space,  whether  this  surface  be  in  a  magnetic 


68  ELECTRICAL  ENGINEERING 

body  or  a  non-magnetic  body  or  in  free  space,  is  defined  as  the 
number  of  lines  of  magnetic  induction  crossing  that  surface,  or  the 
flux  of  magnetic  induction  across  that  surface.  The  unit  of  flux  of 
magnetic  induction  is  called  the  maxwell,  that  is,  1  maxwell  = 
1  c.  g.  s.  line  of  magnetic  induction. 

From  equations  (10)  and  (12)  the  mathematical  expression  for 
the  number  of  lines  of  magnetic  induction  crossing  any  elementary 
area  ds  is 

d  <j)  =(4  77  Jcos  at  +  H  cos  a2)  ds  (14) 

where  J  and  H  are  the  intensity  of  magnetisation  and  the  field 
intensity  respectively  at  ds  and  c^  and  a2  are  the  angles  between  the 
normal  drawn  to  ds  and  the  directions  of  the  intensity  of  mag- 
netisation and  field  intensity  respectively.  The  direction  of  the 
lines  of  induction  through  the  elementary  area  ds  is  taken  as  the 
direction  of  the  vector  which  is  equal  to  the  vector  sum  of  4  77  J  and 
H.  As  a  rule,  in  all  practical  applications  when  the  field  inten- 
sity and  intensity  of  magnetisation  are  due  to  the  same  cause 
(e.  g.,  when  a  piece  of  soft  iron  is  magnetised  by  the  action  of  an 
electric  current),  these  two  quantities  are  in  the  same  direction, 
and  this  expression  may  then  be  written 

d<f)=(4:TTJ+  H)  cos  a.  ds.  (Ha) 

where  a  is  the  common  angle  made  by  J  and  H  with  the  normal 
to  ds;  in  this  case  the  lines  of  force,  lines  of  magnetisation,  and 
lines  of  induction  all  coincide  in  direction. 

From  the  definition  of  lines  of  magnetisation,  there  can  be  no 
lines  of  magnetisation  in  air  or  in  any  other  non-magnetic  sub- 
j  stance.     Hence  in  air  or  in   any   other 

non-magnetic  substance  the  lines  of 
force  and  the  lines  of  induction  are  iden- 
tical, but  this  is  never  the  case  in  a  mag- 
netic or  diamagnetic  substance,  for  when 
Aj  such  a  substance  is  placed  in  a  magnetic 

Fig-  27.  field  it  becomes  magnetised  by  induction 

(see  Article  32)  and  consequently  lines  of  magnetisation,  as  well 
as  lines  of  force,  are  produced  in  the  substance. 

Consider  first  the  lines  of  magnetisation  in  a  single  magnet  and 
the  lines  of  force  due  to  its  poles.  Let  N  be  the  total  number  of 
lines  of  magnetisation  in  this  magnet,  *//„  the  number  of  lines  of 
force  outside  the  magnet,  and  i//t  the  number  of  lines  of  force  inside 
the  magnet.  Outside  the  magnet  the  total  number  of  lines  of  in- 


N 


MAGNETISM  69 

duction  is  then  simply  \jtQ.  Inside  the  magnet,  across  the  section 
A  taken  perpendicular  to  its  axis  at  its  middle  point,  pass  all  the 
lines  of  magnetisation  N  and  the  i/ix  lines  of  force,  the  former  in 
the  direction  S  to  N  and  the  latter  in  the  direction  from  N  to  S. 
Hence  the  total  number  of  lines  of  induction  through  the  magnet 
across  this  area  A  is 

But  since  the  total  number  of  lines  of  magnetisation  is  equal  to 
the  total  number  of  lines  of  force,  we  also  have 


whence  \fj(}  =r  N  —     . 

But  \lfn  is  also  equal  to  the  total  number  of  lines  of  induction  out- 
side the  magnet.  Hence  the  total  number  of  lines  of  induction 
passing  through  a  permanent  magnet  from  its  south  to  its  north 
pole  is  equal  to  the  total  number  of  lines  of  induction  passing  back 
outside  the  magnet  from  its  north  to  its  south  pole.  Therefore  each 
line  of  induction  must  be  a  closed  loop,  part  of  which  lies  inside 
the  magnet  and  part  outside.  The  magnet  may  then  be  looked 


Fig.  28.   Lines  of  Magnetic  Induction  due  to  a  Bar  Magnet. 

upon  as  a  sheath  which  binds  these  lines  of  induction  closely 
together  (Fig.  28);  these  lines  spread  out  from  one  end  of  the  sheath, 
bend  around  and  re-enter  the  sheath  at  the  other  end.  Fig.,  28 
should  be  compared  with  Fig.  29  which  shows  the  lines  of  force  due 
to  a  single  magnet.  It  should  be  noted  that  since  some  of  the  lines 
of  magnetisation  end  in  the  lateral  walls  of  the  magnet,  part  of 
these  lines  of  induction  pass  through  its  lateral  walls.  In  the  case 
of  a  long  slim  magnet,  however,  practically  all  the  lines  of  indue- 


70 


ELECTRICAL   ENGINEERING 


tion  pass  through  its  ends.     In  general,  wherever  there  is  a  mag- 
netic pole  on  the  surface  of  a  magnetic  substance  there  must  also 


Fig.  29.     Magnetic  Lines  of  Force  due  to  a  Bar  Magnet. 

be  a  line  of  induction  passing  through  this  surface;  or,  vice 
versa,  wherever  a  line  of  induction  passes  through  the  surface  of 
a  magnetic  substance  there  must  be  a  pole  on  its  surface. 

Since  each  line  of  induction  due  to  a  single  magnet  forms  a 
closed  loop  it  follows  that  such  a  line  of  induction  will  always  cut 
a  closed  surface  an  even  number  of  times,  as  many  times  in  the 
direction  from  the  outside  to  the  inside  as  in  the  direction  from 
the  inside  to  the  outside.  Hence,  adopting  the  same  convention  as 


Line  of  Induction 


Closed  Surface 


Fig.  30. 

in  the  case  of  lines  of  force,  namely,  that  a  line  entering  a  surface  is 
equivalent  to  a  negative  line  leaving  that  surface,  it  follows  that  the 
algebraic  sum  of  the  lines  of  induction  outward  across  any  closed 
surface  is  always  zero,  even  though  this  surface  encloses  the  pole  of 


MAGNETISM  71 

a  magnet.  The  difference  in  this  respect  between  lines  of  force  and 
lines  of  induction  should  be  carefully  noted;  1,  lines  of  force 
end  on  magnetic  poles,  while  lines  of  induction  pass  through  mag- 
netic poles;  2,  the  lines  of  force  due  to  a  magnet  therefore  have 
ends,  while  the  lines  of  induction  are  closed  curves;  3,  the  alge- 
braic sum  of  the  lines  of  force  outward  across  a  closed  surface  is 
equal  to  4  TT  times  the  algebraic  sum  of  the  magnetic  poles  enclosed 
by  this  surface,  while  the  algebraic  sum  of  the  lines  of  induction 
across  any  closed  surface  is  always  zero. 

48.  Flux  Density.  —  We  have  seen  that  the  number  of  lines  of 
force  per  unit  area  crossing  any  elementary  surface  ds  is  equal  to 
the  component  of  the  intensity  of  the  magnetic  field  normal  to  this 
area  (equation  10).  We  shall  also  find  that  the  number  of  lines 
of  induction  per  unit  area  crossing  any  elementary  surface  plays 
a  very  important  role  in  the  theory  of  magnetism  and  electro- 
magnetism.  When  the  surface  is  taken  normal  to  the  lines  of 
induction  through  it,  the  number  of  these  lines  of  induction  per 
unit  area  crossing  this  surface  is  called  the  density  of  the  lines 
of  induction  at  the  point  occupied  by  this  elementary  surface  or 
simply  the  flux  density  at  that  point,  and  the  direction  of  this 
flux  density  is  defined  as  the  direction  of  the  lines  of  induction  at 
this  surface.  Flux  density  is  therefore  a  vector  quantity,  since 
it  has  both  magnitude  and  direction.  The  symbol  usually  em- 
ployed for  flux  density  is  the  capital  letter  B. 

The  mathematical  expression  for  the  flux  density  in  terms  of 
the  intensity  of  magnetisation  /  and  the  field  intensity  H  when 
both  /  and  H  are  in  the  same  direction  is  therefore 


(15) 

See  equation  (14a). 

When  J  and  H  are  not  in  the  same  direction  4?r  J  and  H  must 
be  added  vectorially  ;  see  equation  (14).  This  can  be  done  by  re- 
solving J  and  H  along  two  mutually  perpendicular  axes,  adding 
the  components  of  4  TT  J  and  H  along  these  respective  axes,  and 
taking  the  square  root  of  the  sum  of  the  squares  ;  see  Article  9. 

The  unit  of  flux  density  on  the  c.  g.  s.  system  is  one  line  of  in- 
duction per  square  centimeter,  or  one  maxwell  per  square  centi- 
meter; this  unit  is  called  the  gauss.  Flux  density  may  also  be 
expressed  as  so  many  c.  g.  s.  lines  per  square  inch,  or  so  many 
thousands  of  c.  g.  s.  lines  per  square  inch.  A  thousand  lines  is 


72  ELECTRICAL   ENGINEERING 

called  a  kilo-line.  Hence  the  following  relations  between  the 
various  units  of  flux  density : 

1  gauss  =  1  maxwell  per  square  centimeter 

1  gauss  =  1  c.  g.  s.  line  per  square  centimeter 

1  line  per  square  inch  =0.15500  gauss 

1  kilo-line  per  square  inch   =155.00  gausses 

From  the  definition  of  flux  density,  the  number  d  <£  of  lines  of 
induction  crossing  any  elementary  surface  ds  may  be  expressed  in 
terms  of  the  flux  density  by  the  formula 

d(j>  =  (Bcosd)  ds  (16) 

where  B  is  the  flux  density  at  ds  and  a  the  angle  between  the 
direction  of  the  lines  of  induction  through  ds  and  the  normal  to 
ds;  this  follows  immediately  from  equation  (14).  Compare  this 
with  the  expression  for  the  number  of  lines  of  force  across  any 
elementary  surface  ds  (equation  10). 

The  above  definition  of  flux  density  and  equations  (14)  and 
(15)  are  applicable  to  any  magnetic  field,  no  matter  how  this  field 
may  be  produced,  whether  by  a  single  permanent  magnet,  by  any 
number  of  magnets,  or  by  an  electric  current.  Moreover,  just  as 
the  separate  sets  of  lines  of  force  due  to  any  number  of  magnets 
may  be  represented  by  a  single  set  of  "  resultant  "  lines  of  force, 
so  the  separate  sets  of  lines  of  induction  due  to  any  number  of  mag- 
nets may  be  represented  by  a  single  set  of  "  resultant  "  lines  of 
induction,  and  these  resultant  lines  of  induction  always  form 
closed  loops,  just  as  the  lines  of  induction  due  to  a  single  magnet 
are  closed  loops. 

For,  let  ds  be  any  elementary  surface  in  the  field,  BI  B2  etc., 
the  flux  densities  at  ds  due  to  the  respective  magnets,  and  a,,  03, 
etc.,  the  angles  between  the  normal  to  ds  and  the  directions  of  the 
lines  of  induction  through  ds  due  to  the  respective  magnets.  Then 
the  number  of  lines  of  induction  across  ds  due  to  the  first  magnet 
is  (Bl  cos  aj)  ds;  the  number  of  lines  of  induction  across  ds  in  the 
same  direction  due  to  the  second  magnet  is  (B2  cos  03)  ds;  etc. 
Hence  the  net  number  of  lines  of  induction  across  ds  in  the  same 
sense  due  to  all  the  magnets  is 

d(j>=  (Bl  cos  a,!  +  B2  cos  a2  +  -   — )  ds  =  (B  cos  a)  ds 

and  the  total  number  of  lines  of  induction  across  any  finite  surface 
is  then 

<j)  =  C  (Bl  cos  a,)  ds  +  C   (B2  cos  a2)  ds  +-    —  =  C  (B  cos  a)  ds 

J  s  J  s  J  s 

where  B  is  the  resultant  or  vector  sum  of  B\  B2  etc.,  and  a  is  the 
angle  between  the  normal  to  ds  and  the  direction  of  the  resultant 


MAGNETISM  73 

flux  density  B.     Hence  we  may  look  upon  the  independent  sets  of 

lines  of  induction  f  (Bi  cos  aO  ds,  C  (B2  cos  a2)  ds,  etc.,  as  combin- 

J  s  J  s 

ing  and  forming  one  set  of  lines  equal  in  number  to  f   (B  cos  a)  ds, 

s  . 
the  direction  of   each  of   these  lines   at  each  point  in  its  path 

coinciding  with  the  direction  of  the  resultant  flux  density  at  that 
point. 

Since  the  net  number  of  lines  of  induction  due  to  a  single  mag- 
net outward  across  any  closed  surface  is  zero  (Article  47,  last  para- 
graph), the  algebraic  sum  of  the  lines  of  induction  due  to  any 
number  of  magnets,  outward  across  any  closed  surface,  must  also 
be  zero,  that  is 
C  (B  cos  a)  ds  —  C  (Bl  cos  a^  ds  +  C  (B2  cos  a2)  ds  +  -  —  =  0 

J  |5|  J  |5|  J  |5| 

since  each  term  of  the  middle  member  of  this  equation  is  zero. 
Hence  the  lines  of  induction  due  to  any  number  of  magnets, 
whether  these  magnets  be  permanent  or  induced,  must  form  closed 
loops,  and  therefore  the  number  of  these  lines  coming  up  to  any 
surface  on  one  side  must  always  be  equal  to  the  number  of  these 
lines  which  leave  the  other  side  of  this  surface.  When  we  come 
to  the  study  of  electric  currents  wre  shall  also  see  that  the  lines  of 
induction  produced  by  an  electric  current  are  also  closed  loops. 
Hence  a  line  of  induction  always  forms  a  closed  loop,  no  matter  how 
it  may  be  produced. 

49.  The  Normal  Components  of  the  Flux  Density  on  the  Two 
Sides  of  any  Surface  are  Equal.  —  The  fact  that  the  number  of  lines 
of  induction  coming  up  to  one  side  of  a  surface  must  equal  the 


Normal 


Fig.  31. 

number  of  these  lines  which  leave  the  other  side,  is  a  very  impor- 
tant one  in  the  theory  of  magnetism,  and  may  be  expressed  mathe- 
matically as  follows.  Let  Bl  and  B3  be  the  flux  densities  on  the 


74  ELECTRICAL  ENGINEERING 

two  sides  of  any  surface  directly  opposite  any  point  P  in  this  sur- 
face, and  let  at  and  cu>  be  the  angles  between  the  direction  of  the 
normal  drawn  through  the  surface  at  this  point  and  the  directions 
of  these  flux  densities.  Let  ds  be  any  elementary  area  of  the  sur- 
face at  P.  Then  the  number  of  lines  of  induction  coming  up  to 
ds  on  one  side  is  (  BI  cos  at)  ds,  and  the  number  of  lines  of  induction 
leaving  ds  on  the  other  side  is  (B2  cos  a2)  ds.  Therefore,  since  the 
number  of  lines  of  induction  coming  up  to  ds  must  equal  the  num- 
ber leaving  ds, 

(Si  cos  a,)  ds  =  (B2  cos  a2)  ds 
or 

#!  cos  a,  =  B2  cos  a2  (17) 

That  is,  the  normal  components  of  the  flux  densities  on  the  two  sides 
of  the  surface  are  equal,  and  this  is  true  even  though  the  surface  is 
the  seat  of  a  magnetic  pole.  When  there  is  no  pole  at  the  surface, 
the  tangential  components  of  the  flux  densities  (Bl  sin  ^  and  B2 
sin  03)  are  also  equal,  but  when  there  is  a  pole  at  the  surface,  the 
tangential  components  will  not  in  general  be  equal.  In  the  latter 
case  the  lines  of  induction  make  an  abrupt  change  in  direction. 
See  Article  54. 

50.  The  Tangential  Components  of  the  Field  Intensity  on  the 
Two  Sides  of  any  Surface  are  Equal.  —  While  lines  of  induction  are 
always  continuous  lines  forming  closed  loops,  and  therefore  always 


B 


Fig.  32. 

pass  through  any  surface  in  their  path,  lines  of  force  originate  at  or 
end  on  magnetic  poles.  Hence  the  normal  components  of  the  field 
intensities  on  the  two  sides  of  a  surface  on  which  there  are  mag- 
netic poles  are  not  equal ;  the  tangential  components  of  these  field 
intensities  are  however  always  equal,  even  though  the  surface  be 
magnetised.  This  may  be  proved  as  follows.  Let  AB  be  the 
normal  through  the  surface  at  any  point  Q  and  let  PI  be  a  point  on 


MAGNETISM  75 

this  normal  just  outside  the  surface  on  one  side  and  P2  a  point  on 
this  normal  just  outside  the  surface  on  the  other  side,  and  let 
QPi  and  QP2  be  equal.  Then  the  field  intensities  at  PI  and  P2 
due  to  any  pole  at  any  point  X  other  than  Q  will  differ  in  mag- 
nitude and  direction  by  an  amount  depending  upon  ,the  distance 
QPl  and  QP2  and  in  the  limit,  when  Pl  and  P2  coincide  with  Qy 
these  field  intensities  due  to  the  pole  at  X  will  be  exactly  equal 
both  in  magnitude  and  direction.  Hence  both  the  normal  and 
the  tangential  components  of  the  field  intensities  at  PI  and  P3 
due  to  a  pole  at  any  other  point  than  Q  will  be  respectively  equal 
when  Pl  and  P2  coincide  with  or  are  infinitely  close  to  Q.  If, 
however,  there  is  a  magnetic  pole  at  Q,  this  pole  will  produce  equal 
and  opposite  field  intensities  at  PI  and  P2,  and  these  intensities 
will  still  be«equal  and  opposite  when  Pl  and  P2  are  infinitely  close  to 
the  surface.  But  since  by  hypothesis  PI  and  P2  are  on  the  normal 
to  the  surface  at  Q,  the  field  intensities  at  these  two  points  due  to 
the  pole  at  Q  will  also  be  normal  to  this  surface  and  therefore  have 
no  tangential  components.  Hence  the  pole  at  Q  has  no  effect 
upon  the  tangential  components  at  PI  and  P2.  Therefore,  since 
the  tangential  components  at  P!  and  P2  due  to  all  the  other  poles 
in  the  field  are  also  equal,  the  resultant  tangential  components  at 
P!  and  P?  due  to  all  the  poles  in  the  field  are  equal.  The  resultant 
normal  components  of  the  field  intensities  at  PI  and  P2  are  how- 
ever not  equal  when  there  is  a  pole  at.Q,  since  the  field  intensity  at 
P!  due  to  the  pole  at  Q  is  opposite  to  the  field  intensity  at  P2  due 
to  this  pole. 

Calling  04  and  o^  the  angles  between  normal  drawn  through  any 
point  of  a  surface  and  the  directions  of  the  field  intensities  Hl  and 
H2  on  the  two  sides  of  this  surface,  we  then  have  that 

H!  sin  ttj  =  H2  sin  a2  (18) 

51,  Conditions  which  must  be  Satisfied  at  every  Surface  in  a 
Magnetic  Field.  —  The  above  deductions  concerning  lines  of  mag- 
netic induction  and  lines  of  force  hold  whether  the  magnets  pro- 
ducing the  field  are  permanent  or  induced.  The  two  surface  con- 
ditions just  deduced  hold  for  every  surface  in  the  field,  and 
consequently  must  hold  at  the  surface  of  any  magnetic  body  placed 
in  the  field.  By  taking  this  fact  into  account  one  can  calculate  in 
certain  simple  cases  the  exact  distribution  of  the  magnetic  poles 
induced  on  the  surface  of  a  magnetic  body  when  placed  in  a 
magnetic  field,  and  also  the  distribution  of  the  lines  of  force  and 
the  lines  of  induction.  (See  J.  J.  Thomson,  El.  of  Elec.  &  Mag. 
p.  257ff .)  These  surface  conditions  may  be  summed  up  : 

1.  The  normal  components  of  the  flux  densities  on  the  two 
sides  of  any  surface  are  always  equal. 

2.  The  tangential  components  of  the  field  intensities  on  the 
two  sides  of  any  surface  are  always  equal. 


76  ELECTRICAL  ENGINEERING 

52.  Induced  Magnetisation. —  We  have  seen  that  when  a  mag- 
netic body  which  itself  is  not  a  magnet  is  placed  in  a  magnetic 
field,  this  body  becomes  a  magnet;  this  phenomenon  is  de- 
scribed by  saying  that  the  body  becomes  magnetised  by  in- 
duction. To  fix  ideas,  let  this  field  be  that  in  the  vicinity  of  the 
north  pole  of  a  bar  magnet,  and  let  the  magnetic  body  be  a  soft  iron 


B 


Fig.  33. 

rod  placed  as  shown.  As  we  have  already  seen,  the  end  A  of 
this  rod  becomes  a  south  pole  and  the  end  B  a  north  pole.  Con- 
sequently inside  the  rod  A  B  there  will  be  (1)  a  field  of  force  due 
to  the  original  field  of  the  bar  magnet  in  the  direction  from  A 
to  B  and  (2)  a  field  of  force  due  to  the  magnetic  poles  "  in- 
duced "  on  the  rod,  which  field  will  be  in  the  direction  from  B  to 
A.  Let  H  be  the  numerical  value  of  field  intensity  at  any  point 
P  inside  the  rod  due  to  the  original  magnetic  field,  H'  the 
numerical  value  of  the  field  intensity  at  this  point  due  to  the  poles 
induced  on  the  rod,  and  J  the  numerical  value  of  the  intensity  of 
magnetisation  of  the  rod  at  this  point.  The  induced  intensity  of 
magnetisation  /  may  be  considered  as  produced  by  the  original 
field  H.  The  intensity  H'  which  is  due  to  the  induced  poles  and 
which  therefore  is  approximately  in  the  opposite  direction  to  the 
original  field  H,  would  of  itself  tend  to  magnetise  a  body  in  the 
opposite  direction,  and  is  therefore  called  the  "  demagnetis- 
ing force  "  due  to  the  ends  of  the  rod.  Experiment  shows  that 
this  demagnetising  force  H'  is  always  less  than  the  magnetis- 
ing force  H  but  is  not  negligible  unless  the  field  in  which  -the  rod 
is  placed  is  uniform  and  the  rod  itself  is  very  long.  The  resultant 
field  intensity  at  P  will  then  be  the  vector  difference 


Hr  =  H  -  H' 

When  H  and  H'  are  in  exactly  opposite  directions  the  resultant 
field  intensity  Hr  is  the  arithmetical  difference  between  H  and 

#'or  Ur  =  E-Rr 

This  condition  is  approximately  realized  in  the  case  of  a  long, 


MAGNETISM  77 

slim  bar  placed  in  a  uniform  magnetic  field  parallel  to  the  di- 
rection of  the  field.  Experiment  also  shows  that  when  a  body 
which  of  itself  is  not  a  magnet  (e.g.,  the  soft  iron  rod  we  are 
considering)  is  placed  in  a  magnetic  field,  the  intensity  of  mag- 
netisation produced  at  any  point  in  the  body  is  in  general  in  the 
direction  of  the  resultant  field  intensity.  (Certain  exceptions  will 
be  noted  later,  Article  57).  Consequently,  across  any  elementary 
surface  ds  drawn  normal  to  the  direction  of  the  resultant  field  in- 
tensity there  will  be  Hr  ds  lines  of  force  and  4  TT  J  ds  lines  of  mag- 
netisation, both  as  a  rule  in  the  same  direction,  namely  from  A  to  B 
(see  equations  10  and  12).  Hence  the  total  number  of  lines  of 
induction  across  this  area  will  be  (see  equation  14a) 

d<f>  =  (Hr  +  4iTJ)  ds 

and  therefore  the  flux  density  at  the  point  P,  i.e.,  the  number  of 
lines  of  induction  per  unit  area  normal  to  the  direction  of  these 
lines,  will  be  ,  , 

(19) 


53.  Magnetic  Permeability.  —  The  resultant  intensity  of  mag- 
netisation J  produced  in  a  piece  of  iron  or  steel  when  it  is  placed 
in  a  magnetic  field  is  in  general  many  times  greater  than  the 
original  intensity  of  the  field,  and  therefore  the  number  of  lines 
of  induction  through  the  space  occupied  by  this  piece  of  iron  or 
steel  is  greatly  increased  by  its  presence,*  although  the  number 
of  lines  of  force  through  this  space  is  in  general  decreased  (the 
latter  is  always  true  except  when  there  are  no  poles  produced  on 
the  surface  of  the  body  which  is  magnetised  by  induction  —  we 
shall  see  later  when  we  come  to  the  study  of  electric  currents  how  a 
body  may  be  magnetised  without  producing  any  poles  on  its  surface)  . 

Hence  iron  or  steel,  or  in  fact  any  magnetic  substance,  is  said 
to  be  more  "  permeable  "  to  lines  of  induction  than  a  non-mag- 
netic body  (diamagnetic  bodies  are  less  permeable),  and  the  ratio 

*The  fact  that  when  a  magnetic  body  is  placed  in  a  magnetic  field  the 
number  of  lines  of  induction  through  the  space  occupied  by  the  body  is  in- 
creased, is  frequently  described  by  saying  that  the  lines  of  induction  tend 
to  crowd  into  a  magnetic  body  when  placed  in  a  magnetic  field.  In  general, 
however,  the  presence  of  such  a  substance  in  a  magnetic  field  not  only 
causes  the  original  lines  of  induction  to  crowd  into  the  substance,  but  also 
gives  rise  to  an  additional  number  of  these  lines.  This  is  particularly  true 
in  the  case  of  an  iron  rod  placed  in  the  magnetic  field  produced  by  an  elec- 
tric current  flowing  in  a  coil  of  wire. 


78  ELECTRICAL  ENGINEERING 

of  the  resultant  flux  density  at  any  point  of  the  body  to  the  result- 
ant field  intensity  is  called  the  permeability  of  the  body  at  that 
point,  and  is  usually  represented  by  the  symbol  ^.  That  is 

._?  (20) 

"  ~ff, 

where  Hr  is  the  resultant  field  intensity,  which  in  case  just  con- 
sidered is  equal  to  H-H'.  Since  there  can  be  no  lines  of  mag- 
netisation in  a  non-magnetic  body,  the  flux  density  and  field  in- 
tensity in  such  body  are  numerically  equal,  and  therefore  for  all 
non-magnetic  bodies  p  =  1.  For  magnetic  bodies  p.  is  always  greater 
than  ]  and  for  diamagnetic  bodies  less  than  1.  Strictly  speaking, 
the  permeability  is  unity  only  for  air,  since  all  bodies  are  slightly 
magnetic  or  diamagnetic  with  respect  to  air.  However,  p  is 
practically  unity  for  all  substances  other  than  iron,  steel,  nickel, 
cobalt  and  bismuth;  for  the  last  p  is  less  than  unity  and  for  the 
rest  greater  than  unity. 

The  permeability  of  any  of  these  substances  is  not  a  constant, 
but  depends  upon  the  intensity  of  the  resultant  magnetic  field, 
and  also  upon  its  previous  history,  whether  it  is  already  a  magnet 
before  being  placed  in  the  field  of  force  and  upon  how  the 
field  of  force  inducing  the  magnetisation  is  established.  In 
fact,  the  permeability  may  be  negative,  that  is,  the  resultant  flux 
density  may  be  in  the  opposite  direction  to  the  resultant  field 
intensity.  This  is  always  true  of  a  single  permanent  magnet  by 
itself.  In  practice,  however,  the  permeability  of  a  body  is  taken 
to  mean  the  ratio  of  the  flux  density  to  the  resultant  field  intensity 
when  the  body  is  originally  unmagnetised  and  then  placed  in  a  field 
which  is  continuously  increased  from  zero  to  its  final  value. 

64.  Refraction  of  the  Lines  of  Induction  at  the  Surface  of 
Separation  of  Two  Bodies  of  Different  Permeabilities.  —  It  can 
readily  be  shown,  by  making  use  of  the  surface  conditions  given 
in  Article  51,  that  wherever  a  line  of  induction  crosses  the  surface 
of  separation  between  two  bodies  of  different  permeabilities,  this 
line  is  refracted  toward  the  normal  at  this  surface  in  the  body 
of  lesser  permeability.  Let  the  permeabilities  of  the  two  bodies 
directly  opposite  any  point  Q  in  the  surface  separating  them  be 
/*i  and  /Kg  respectively  :  let  HI  and  H2  be  the  field  intensities  in  the 
two  bodies  respectively  at  points  infinitely  close  to  Q  on  the  two 
sides  of  the  surface;  let  BI  and  B2  be  the  flux  densities  at  these  two 
points  and  ^  and  o^  the  angles  between  the  normal  drawn  through 
the  surface  at  Q  and  the  directions  on  the  two  sides  of  the  surface 
of  the  Tine  of  induction  through  Q.  The  line  of  force  through 


MAGNETISM 


79 


Q  will  coincide  in  direction  with  the  line 
of  induction  (provided  the  bodies  are  not 
crystals  and  are  magnetised  solely  by 
induction),  hence  at  and  a*  will  also  be 
the  angles  between  the  normal  at  Q  and 
the  directions  of  the  lines  of  force  on  the 
two  sides  of  the  surface.  Hence  from  the 
surface  conditions  given  in  Article  51, 


Hence 


But 


m 

#!  cos  ax  =  B2  cos  02 
H i  sin  0*1  =  H2  sin  ctg 


f  to 


H2 

—- tan  a2. 


Hl  and  B2  =  ^  H2.     Whence 


Fig.  34. 


tan 


=  —  tan  a2. 


(21) 


Hence  if  /^  is  less  than  ^  then  at  is  less  than  a^;  that  is,  the  line 
of  induction  is  bent  toward  the  normal  in  the  body  of  lesser  per- 
meability. For  example,  the  permeability  of  soft  iron  is  about 
3000  under  ordinary  conditions.  Hence  a  line  of  induction  coming 
up  to  the  surface  inside  the  iron  at  an  angle  of  45°  say,  comes  out 
into  the  air  at  an  angle  of  a=  tan'1  sinnr  w^n  the  normal,  that 
is,  comes  out  into  the  air  practically  at  right  angles  to  the  surface. 
Therefore,  when  a  piece  of  soft  iron  is  placed  in  a  magnetic  field  in 
air  practically  all  the  lines  of  induction  which  pass  through  it  enter 
and  leave  its  surface  approximately  at  right  angles  to  the  surface. 
55.  Value  of  the  Pole  Strength  per  Unit  Area  Induced  on  the 
Surface  of  Separation  of  Two  Bodies  of  Different  Permeabilities. — 
From  the  surface  condition  that  the  number  of  lines  of  induction 
coming  up  to  the  surface  is  equal  to  the  number  of  lines  leaving 
the  surface,  can  also  be  deduced  an  expression  for  the  net  pole 
strength  per  unit  area  induced  on  the  surface  of  separation  be- 
tween the  two  bodies.  Let  ds  (Fig.  35)  be  an  elementary  area  in 
this  surface  at  Q  and  draw  about  ds  a  closed  cylinder  having  the 
cross  section  ds  and  its  ends  infinitely  close  to  ds.  Applying 
Gauss's  Theorem  (Art.  44)  to  this  closed  cylinder,  we  have 


4  TT  o~  ds  =  HI  cos  ax  ds  —  H2  cos  a,2  ds 


whence 


=  7-=    cos  ai- 


H, 

H, 


cos 


where  cr  is  the  pole  strength  per  unit  area  on  ds.  But  since  the 
number  of  lines  of  induction  coming  up  to  ds  is  equal  to  the  num- 
ber of  lines  of  induction  leaving  ds,  we  also  have 

BI  cos  (LI  =  B2  cos  d2 
or,  since  Bl=plHl  and  B2=p2H2 


80 


whi 


ELECTRICAL  ENGINEERING 
substituted  in  the  above  equation,  gives 

o.=J^LcosoLr1_^1~i 

47T  frj 


(22) 


Fig.  35 


Hence  if  the  line  of  induction  at  Q  passes  from  a  medium  of  low 
to  one  of  high  permeability  (i.e.,  t^>t^j  the  pole  induced  on  the 
surface  is  negative,  while  if  the  line  of  induction  passes  from  a  med- 
ium of  high  to  one  of  low  permeability  (i.e.,  ^  <&)  the  pole  induced 
on  the  surface  is  positive.  This  explains  the  fact,  noted  in  Article 
29,  that  the  direction  of  the  force  produced  on  an  originally  unmag- 
netised  body  placed  near  a  magnet  depends  upon  the  nature  of  the 
medium  between  the  body  and  the  magnet,  and  that  whether  this 
force  is  an  attraction  or  a  repulsion  depends  upon  the  relative  per- 
meabilities of  the  body  and  the  surrounding  medium.  When  ^ 
and  /*2  are  both  different  from  unity  there  will  be  a  pole  induced 
on  each  of  the  substances  in  contact;  equation  (22)  gives  the  alge- 
braic sum  of  these  pole  strengths  per  unit  area. 

56.  Field  Intensity  at  any  Point  in  Magnetic  Medium  of  Constant 
Permeability  Completely  Surrounding  a  Point-Pole  and  Filling  all 
Space.  — An  important  relation  in  the  theory  of  magnetism  is  that 
the  resultant  field  intensity  at  any  point  in  a  magnetic  medium 
of  constant  permeability  /A  completely  surrounding  a  point-pole 

of   strength  m  isH0= — 2,  where  r  is  the  distance  of   the   point 

from  the  pole.  This  is  equivalent  to  stating  that  the  pole 
m  induces  on  the  surface  of  the  medium  in  contact  with  it 
a  pole  of  opposite  sign,  the  numerical  strength  of  which  is  equal 


to  ( 1  —  I  m,  for  then  the  resultant  field  intensity  will  be 

"      ™  ™ 


(23) 


MAGNETISM  81 

To  prove  this,  consider  a  magnetic  filament  of  infinite  length 
and  infinitesimal  cross  section,  and  let  the  poles  of  this  filament 
have  the  constant  strengths  m  and  —  m.  We  may  imagine  the  pole 
—  m  at  infinity  and  therefore  producing  no  field  intensity  at  any 
point  in  the  vicinity  of  m.  Again,  since  the  filament  is  assumed 
to  have  infinitesimal  cross  section,  we  may  neglect  the  non- 
uniformity  produced  by  it  in  the  medium  surrounding  m.  The 
lines  of  force  due  to  the  pole  m  will  therefore  be  equally  spaced 
lines  radiating  out  from  m  in  all  directions.  When  the  pole  is 
surrounded  by  a  non-magnetic  medium,  i.e.,  one  having  unit  per- 
meability, the  lines  of  induction  in  the  surrounding  medium  will 
coincide  with  these  lines  of  force;  the  lines  of  induction  come  up  to 
the  pole  through  the  magnetic  filament,  as  lines  of  magnetisation, 
and  then  radiate  out  from  the  pole,  as  lines  of  force,  into  the  sur- 
rounding medium.  Since  the  pole  m  is  constant  by  hypothesis, 
the  intensity  of  magnetisation  of  the  filament  must  also  be  con- 
stant; whence  the  number  of  lines  of  magnetisation,  4irm,  in  the 
filament  is  constant,  and  since  the  filament  has  an  infinitesimal 
cross  section,  the  lines  of  force  inside  the  filament  will  be  negligible 
in  comparison  with  the  number  of  lines  of  magnetisation.  Hence 
the  number  of  lines  of  induction  coming  up  to  the  pole  m  through 
the  filament  must  also  be  constant  and  equal  to  4  TT  m;  con- 
sequently the  number  of  lines  of  induction  radiating  out  into  the 
surrounding  medium  is  4  TT  m  independent  of  the  nature  of  the 
medium. 

When  the  pole  is  surrounded  by  a  non-magnetic  medium,  the 

field  intensity  at  any  point  isH  =  — ,  and  since  the  medium  is  non- 
magnetic the  flux  density  is  also  B  =  -^.  Since  the  lines  of 

induction  are  the  same  whether  the  surrounding  medium  is 
magnetic  or  non-magnetic,  the  flux  density  at  any  point  in  any 
medium  whatever  completely  surrounding  a  point-pole  of  strength 

m  is  B  =—.  Consequently,  when  the  medium  has  a  permeability  /x, 
the  field  intensity  at  any  point  in  it  due  to  the  pole  m  must  be 

H— — = — „.     The  number  of  lines  of  force  outward  across  the 
p.      pr2 

surface  of  any  sphere  surrounding  m  is  therefore  -     — ;  but  by 

Gauss's  Theorem  this  must  be  equal  to  4  TT  times  the  algebraic 
sum  of  the  poles  inside  this  sphere;  hence  calling  mi  the  value 
of  the  strength  of  the  pole  induced  by  m  on  the  surface  of  the 

medium  in  contact  with  it,  we  have,  4  TT  (m  +  mi)  ==—  —  and  there- 
fore W;  =  —  (  1 J  m. 

\       M 
From  the  fact  that  the  resultant  field  intensity  at  any  point  in  a 


82  ELECTRICAL   ENGINEERING 

medium  of  constant  permeability  completely  surrounding  a  point- 
pole  of  strength  m  is  — 3,  it  follows  that  the  resultant  force  of  repul- 
sion on  a  second  point-pole  of  strength  mf  placed  in  the  medium  at 
any  point  is  /  =  — — ,  where  r  is  the  distance  between  the  poles  and  ^ 

is  the  permeability  of  the  medium.  It  should  be  noted  that  this 
is  the  resultant  force  acting  on  m'  due  to  both  the  pole  m  and  the 

pole  —  1 1  — -)m  induced  by  m  on  the  surface  of  the  medium  in  con- 

V       A*/  mmf 

tact  with  it.     The  force  due  to  m  alone  is  — - — and  is  independent 

of  the  nature  of  the  surrounding  medium. 

The  above  deductions  hold  only  when  the  poles  are  completely 
surrounded 'by  the  medium  of  constant  permeability  /x.  In  any 
actual  case  this  can  never  be  true,  since  a  magnetic  pole  can  exist 
only  where  there  is  a  surface  of  separation  between  two  substances 
of  different  permeabilities. 

57.  Magnetic  Hysteresis.  —  As  noted  in  Article  53,  the  flux 
density  produced  in  a  given  piece  of  iron  or  other  magnetic  sub- 
stance by  a  given  field  intensity  depends  upon  the  previous  his- 
tory of  the  sample.  To  make  this  fact  clearer,  consider  the  special 
case  of  an  originally  unmagnetised  rod  of  soft  iron  placed  in  a  uni- 
form magnetic  field  with  its  axis  parallel  to  the  direction  of  the  field, 
and  let  this  field  be  gradually  increased  from  zero  up  to  some  maxi- 
mum value  Hm  and  then  decreased  to  zero,  then  increased  in  this 
reversed  direction  to  an  equal  negative  maximum  value  —  Hm,  de- 
creased to  zero  again  and  then  again  increased  to  the  same  maxi- 
mum value  Hm  in  the  original  direction.  It  is  found  by  experi- 
ment that  the  relation  between  the  flux  density  and  the  field 
intensity  during  the  various  steps  in  this  process  may  be  repre- 
sented by  a  curve  of  the  form  shown  in  Fig.  36.  (The  experi- 
mental method  of  determining  such  a  curve  is  described  in  Chapter 
IV.)  From  this  curve  it  is  seen  that  at  first,  when  the  field 
intensity  is  small,  the  flux  density  increases  relatively  slowly  as  the 
field  intensity  increases.  When  the  field  intensity  is  increased 
further,  the  flux  density  increases  very  rapidly;  when  the  field  in- 
tensity becomes  still  greater,  the  flux  density  increases  more  and 
more  slowly,  and  finally  any  further  increase  in  the  field  intensity 
produces  only  a  comparatively  slight  change  in  the  flux  density. 
When  the  field  intensity  is  now  reduced  the  flux  density  instead 
of  returning  to  the  same  values  it  had  for  the  increasing  values  of 
the  field  intensity,  decreases  less  rapidly  than  it  increased,  that  is, 


MAGNETISM 


83 


the  decreasing  values  of  the  flux  density  lag  behind  the  values 
corresponding  to  an  increasing  field  intensity.     This  phenomenon 


-II 


-12    -10     - 


-B 


B 


-14000 


II 


+  10    + 


Fig.  36. 

has  therefore  been  given  the  name  hysteresis,  a  Greek  word  meaning 
"  lagging  behind." 

When  the  field  intensity  is  reduced  to  zero;  the  flux  density  still 
has  a  considerable  value;  this  value  is  called  the  "  remanent 
magnetism  "  of  the  sample.  To  reduce  the  flux  density  to  zero; 
the  field  intensity  has  to  be  reversed  and  increased  to  a  value 
corresponding  to  the  abscissa  of  the  point  where  the  left-hand 
branch  of  the  curve  cuts  the  axis  of  field  intensities.  This  value 
of  the  field  intensity  is  called  the  "  coercive  force."  When 
the  field  intensity  is  still  further  increased  in  the  negative 
direction  the  flux  density  reverses  in  direction,  increasing  very 
rapidly  at  first  and  then  more  slowly.  When  the  field  intensity 
reaches  the  same  value  in  the  negative  direction  as  its  maximum 


84  ELECTRICAL  ENGINEERING 

value  in  the  positive  direction,  the  flux  density  likewise  reaches  a 
maximum  in  the  negative  direction  equal  to  the  maximum  value 
it  had  in  the  positive  direction.  When  the  field  intensity  is  now 
reduced  to  zero  and  then  increased  again  in  the  positive  direction 
to  its  original  maximum  value,  the  flux  density  passes  through  the 
series  of  values  represented  by  the  right-hand  branch  of  the  curve, 
which  is  perfectly  symmetrical  with  the  left-hand  branch.  The 
closed  curve  formed  by  these  two  branches  is  called  the  "  hysteresis 
loop." 

It  should  be  noted  that  when  the  sample  is  originally  un- 
magnetised  the  curve  giving  the  relation  between  the  flux  den- 
sity and  the  field  intensity  when  the  field  intensity  is  first  increased 
from  zero  up  to  the  maximum  value,  does  not  coincide  with  either 
branch  of  the  hysteresis  loop,  but  is  a  curve  which  in  general  lies 
between  the  two  branches  of  the  loop.  After  the  completion  of  a 
cycle  of  changes  in  the  field  intensity  from  a  positive  maximum  to 
an  equal  negative  maximum  and  then  back  again  to  the  positive 
maximum,  the  field  intensity  may  then  be  reversed  back  and  forth 
any  number  of  times  between  these  equal  positive  and  negative 
maximum  values,  and  the  relation  between  the  flux  density  and 
the  field  intensity  will  be  the  same  for  each  cycle  of  changes  in  the 
field  intensity  as  for  the  first  cycle.  If  the  iron  is  not  originally 
unmagnetised,  the  first  hysteresis  loop  will  be  shifted  above  or 
below  the  axis  of  field  intensities,  but  after  a  number  of  reversals 
of  the  .field  intensity  between  given  positive  and  negative  values, 
the  loop  will  become  practically  symmetrical  with  this  axis,  par- 
ticularly if  the  iron  is  continually  jarred.  In  the  armatures  of 
electrical  machines  and  the  cores  of  transformers,  in  which  the  field 
intensity  reverses  a  large  number  of  times  every  second  and  the 
iron  is  continually  jarred,  the  relation  between  flux  density  and 
field  intensity,  after  a  short  interval  of  time,  is  represented  by 
a  symmetrical  loop  of  the  form  shown  in  Fig.  36. 

The  area  enclosed  by  the  hysteresis  loop  depends  upon  the 
maximum  value  of  the  flux  density  reached  during  the  cycle,  but 
the  general  shape  remains  about  the  same.  Fig.  37  shows  a  series 
of  loops  corresponding  to  various  values  of  the  maximum  flux 
density.  The  area  of  the  loop  is  also  different  for  various  kinds  of 
iron  or  steel.  As  we  shall  see  when  we  come  to  the  study  of  electric 
currents  (Chapter  IV),  the  area  of  this  loop  represents  a  certain 
amount  of  energy  dissipated  as  heat  energy  in  the  iron;  in  fact,  the 


MAGNETISM 


85 


ergs  of  heat  energy  dissipated  per  cycle  is  equal  to  —  times  the 

4  TT 

area  of  this  loop,  when  both  the  flux  density  and  the  field  in- 
tensity are  plotted  to  the  same  scale. 

B 


+  50 


Fig.  37. 

On  account  of  this  energy  loss  due  to  hysteresis,  it  is  desirable 
to  have  all  parts  of  the  magnetic  circuit  of  an  electric  machine  in 
which  there  is  a  varying  magnetic  flux  made  of  iron  or  steel  in 
which  this  hysteresis  loss  is  a  minimum.  It  has  been  found  that  in 
iron  which  contains  about  three  per  cent  silicon  this  hysteresis 
loss  is  about  half  what  it  is  in  the  best  grade  of  low  carbon  steel. 
This  so-called  "  silicon  steel  "  is  now  being  extensively  used  in  the 
construction  of  electric  machines,  particularly  transformers. 

An  examination  of  this  hysteresis  loop  also  makes  clear  how  a 
bar  of  steel  may  be  permanently  magnetised  by  placing  it  in  a 
magnetic  field.  For,  when  the  bar  is  removed  from  the  field  it 
retains  an  intensity  of  magnetisation  approximately  equal  to  the 
flux  density  where  the  hysteresis  loop  cuts  the  axis  of  flux  densities, 
that  is,  an  intensity  of  magnetisation  approximately  equal  to  the 
remanent  magnetism.  Experiment  shows  that  a  hard  steel  bar 


86 


ELECTRICAL   ENGINEERING 


thus  magnetised  may  be  handled  with  comparative  roughness 
without  reducing  to  any  considerable  extent  the  strength  of 
its  poles,  but  in  the  case  of  a  soft  iron  bar  even  the  slightest  jar 
will  cause  it  to  lose  its  magnetism  almost  entirely.  The  property 
possessed  by  a  hard  steel  bar  of  retaining  its  magnetisation  is 
called  its  "  retentiveness." 

58.  Normal  B-H  Curves.  — Magnetic  Saturation.  — The  curve 
giving  the  relation  between  the  flux  density  and  the  resultant  field 
intensity  when  the  latter  is  increased  from  zero  up  to  successively 


40  50  ~60~ 

H  in  C.Q.S.  Electromagnetic  Units 

Fig.  38.* 

greater  values  is  called  the  "  normal  B-H  curve."  In  Fig.  38  are 
given  the  normal  B-H  curves  for  cast  iron,  cast  steel  and  annealed 
sheet  steel  (low  carbon)  such  as  is  ordinarily  used  in  the  construc- 
tion of  electric  machines.  In  Fig.  39  are  given  the  corresponding 
curves  showing  the  relation  between  the  intensity  of  magnetisation 
and  the  field  intensity  (calculated  from  the  B-H  curves  by  equation 
18),  and  in  Fig.  40  the  corresponding  permeability  curves  (calcu- 
lated from  the  B-H  curves  by  equation  19).  It  will  be  noted 
that  the  intensity  of  magnetisation  corresponding  to  values  of  the 
flux  density  above  the  sharp  bend  or  knee  in  the  B-H  curves, 

^Standard  curves  used  by  General  Electric  Co. 


MAGNETISM 


87 


Fig    39. 


B 

\ 

\ 

•^x 

12000 

\ 

\ 

^^ 

^ 

\ 

\ 

"**^~ 

•^^ 

^ 

s 

\ 

She 

;t  St 

-et^ 

\ 

\ 

Cas 

•  Ste 

el 

^ 

Xw 

8000 

\ 

^ 

Nj 

1 

\ 

\ 

\ 

\ 

/ 
/ 

^ 

---" 

'" 

f 

t  In 

>n 

/ 

/ 

/ 

^ 

~~" 

^^ 

2000 

— 

| 

/ 

/ 

800      1200      1600      2000      2400      2800      3200      3600     40UU 

Fig.  40 


88  ELECTRICAL  ENGINEERING 

increases  very  slowly  and  soon  becomes  practically  constant 
independent  of  the  value  of  the  field  intensity.  In  fact,  experi- 
ment shows  that  it  is  impossible  to  produce  an  intensity  of 
magnetisation  in  a  given  substance  greater  than  a  certain  defi- 
nite value,  which  is  different  for  different  substances.  When 
a  magnetic  substance  is  thus  magnetised  to  its  maximum  in- 
tensity of  magnetisation  it  is  said  to  be  "  saturated."  Such  a 
substance  is  practically  saturated  for  any  value  of  the  field  in- 
tensity well  above  the  knee  of  the  B-H  curve. 

It  should  be  noted  that  the  B-H  curves  for  iron  and  steel  de- 
pend to  a  very  great  extent  upon  the  physical  structure  and  chemi- 
cal constitution  of  the  sample,  and  the  heat  treatment  to  which 
it  has  been  subjected.  It  has  also  been  recently  discovered  that 
when  iron  is  annealed  in  an  alternating  magnetic  field,  the  per- 
meability is  increased  in  certain  cases  as  much  as  50  per  cent.  The 
B-H  curves  of  two  samples  taken  from  the  same  lot  of  material 
may  even  differ  considerably.  The  curves  also  depend  upon  the 
temperature  of  the  sample  at  the  time  the  observations  are  taken, 
though  the  variation  due  to  ordinary  changes  of  temperature  is 
slight.  For  very  high  temperatures,  however,  all  magnetic  sub- 
stances become  practically  non-magnetic.  This  temperature  cor- 
responds to  the  major  recalescence  point,  which  is  about  750 
degrees  for  steel  of  the  quality  used  in  armature  and  transformer 
punchings.  When  iron  is  kept  continuously  at  a  moderately  high 
temperature  (100°C),  the  hysteresis  loop  also  gradually  increases 
in  size,  and  therefore  the  energy  loss  in  the  magnetic  circuits  of 
electric  machines  due  to  "  hysteresis  "  increases  with  time.  This 
effect  is  called  "  aging."  There  is  practically  no  aging  of  silicon 
steel. 

59.  'Magnetic  Potential.  —  Whenever  the  position  of  a  mag- 
netic pole  with  respect  to  another  pole  is  "changed,  work  must  in 
general  be  done,  either  by  the  poles  themselves  or  by  the  external 
agent  which  produces  this  change  in  the  relative  position  of  the  two 
poles.  For,  every  magnetic  pole  produces  a  force  on  every  other 
pole,  and  consequently  when  one  pole  moves  with  respect  to  the 
other  there  is  a  force  and  a  displacement,  and,  by  definition,  work 
is  equal  to  the  product  of  the  displacement  by  the  component  of  the 
force  in  the  direction  of  the  displacement  (Article  21).  The  total 
work  which  two  permanent  poles  of  like  sign  at  a  given  distance 
apart  can  do  on  each  other,  is  the  work  done  when  the  two  poles 


MAGNETISM  89 

move  from  their  original  positions  to  an  infinite  distance  apart. 
This  amount  of  work  may  be  looked  upon  as  the  relative  potential 
energy  of  the  two  poles,  since  it  is  the  work  they  are  capable  of,  or 
have  the  "  potentiality  "  of,  doing  on  each  other.  Similarly,  the 
work  done  by  any  number  of  magnetic  poles  in  moving  a  unit 
north  point-pole  from  a  given  point  to  an  infinite  distance  may  be 
looked  upon  as  the  relative  potential  energy  with  respect  to  these 
poles  of  a  unit  north  pole  located  at  this  point.  This  quantity 
is  called  simply  the  potential  at  the  point;  i.e.,  the  potential 
at  any  point  in  a  magnet  field  is  the  work  done  by  the  poles 
producing  the  field  in  moving  a  unit  north  point-pole' from  that  point 
to  infinity,  provided  these  poles  remain  constant  in  strength  and 
their  relative  positions  remain  unaltered.  Potential  is  then  work 
divided  by  pole  strength,  and  since  both  work  and  pole  strength 
are  scalar  quantities,  potential  is  also  a  scalar  quantity.  Hence 
potentials  may  be  added  algebraically. 

The  potential  at  any  point  P  due  to  a  point-pole  of  strength 
m  at  a  distance  r  away  can  be  readily  calculated.     Consider  any 


Fig.   41. 

point  Q  on  the  line  drawn  through  the  pole  m  and  the  point  P; 
let  this  point  be  at  a  distance  x  from  m.  Then  the  force  which  would 

act  on  a  unit  north  pole  at  Q  is  —  in  the  direction  from  m  to  Q. 

x2 

When  the  unit  pole  moves  a  distance  dx  along  this  line,  the  work 
done  by  m  on  it  is  — — .  Consequently,  when  the  unit  pole 
moves  along  this  line  from  P  to  infinity  the  work  done  is 


90  ELECTRICAL  ENGINEERING 


Joo  r         noo ' 

mdx  .1  m 

--=m\  -  -       =~ 
r     X2  Xjr         r 


Hence  the  potential  at  any  point  due  to  a  point-pole  m  at  a  dis- 
tance r  away  is 

y=™  (24) 

r 

When  m  is  a  south  pole,  —  is  negative,  which  means  that  the 

r 

unit  pole  does  work  against  the  force  due  to  m,  instead  of  this 
force  doing  work  on  the  unit  pole.  It  can  readily  be  shown  that 
the  work  done  by  the  pole  m  on  the  unit  north  pole,  when  the  latter 
moves  from  the  point  P  to  infinity,  is  independent  of  the  path 
over  which  the  unit  pole  moves.  For,  the  work  done  on  the  unit 
pole  when  it  moves  a  distance  dl  in  any  direction  making  an  angle 

0  with  the  line  from  mto  Q  is  —  (dl  cos  0).     But  dl  cos  9  is  equal 

to  the  projection  of  dl  on  the  line  from  m  to  Q.  Calling  this 
projection  dx,  we  have  that  the  work  corresponding  to  the  dis- 
placement dl  is  ,  which  integrated  between  the  limits  x=r 

and  £  =  oo;  gives  the  same  value  of  the  potential  as  before. 
Hence  the  potential  at  any  point  due  to  a  point-pole  of  strength 
m  depends  only  on  the  position  of  this  point  with  respect  to  m 
and  the  value  of  m. 

The  potential  at  any  point  due  to  any  number  of  point-poles 
ml  m2  etc.,  at  the  distances  rl  r2  etc.,  from  this  point  is  then  the 
algebraic  sum 

V=m1+m,+ (24fl) 

r,      r2 

and  the  potential  at  any  point  due  to  any  magnetised  surface  is 

(T  ds 


•• 


~  '  (246) 


where  ds  is  any  elementary  area  in  this  surface,  <r  the  pole  strength 
per  unit  area  at  ds,  and  r  the  distance  of  ds  from  the  point  con- 
sidered, and  J  s  indicates  the  sum  of  the  expressions  ^  for  ail 

r 
the  elements  into  which  the  surfaces  are  divided. 


MAGNETISM  91 

60.   Difference    of   Magnetic    Potential.  — The    difference    of 
potential  between  any  two  points  1  and  2  in  a  magnetic  field;  or 


Fig.  42. 

specifically,  the  drop  in  magnetic  potential  from  the  point  1  to 
the  point  2,  is  equal  to  the  work  that  would  be  done  by  the  agents 
producing  the  field  were  a  unit  north  point-pole  moved  from  1  to 
2.  Let  Vl  and  72  be  the  potentials  at  the  two  points  respectively, 
then-  the  drop  of  potential  from  1  to  2  is  V^—  V2.  Let  dl  repre- 
sent an  elementary  length  in  the  path  over  which  the  unit  pole 
moves,  H  the  field  intensity  at  dl  and  0  the  angle  between  dl 
and  the  direction  of  the  field  intensity  at  dl;  then  the  work  that 
would  be  done  by  the  agents  producing  the  field  on  a  unit  north 

point-pole  when  the  latter  moves  from  1  to  2  is    f  2  (H  cos  9)  dl 

J  i 

where 

V,-VZ=  C  (H  cos  9)  dl  (25) 

J  i 

When  Y2  is  greater  than  Vl  the  drop  of  potential  from  1 
to  2  is  negative,  that  is  the  integral  f -  ( H  cos  9)  dl  is  a  negative 

quantity.  A  negative  drop  of  potential  in  any  direction  is  equiva- 
lent to  a  positive  rise  in  that  direction. 

When  the  magnetic  field  is  due  solely  to  magnetic  poles  the 
drop  of  magnetic  potential  around  any  closed  path  is  zero,  since 
the  magnetic  potential  at  any  point  due  to  any  number  of  poles 
can  have  but  a  single  value.  This,  however,  is  not  true  when 
the  path  links  an  electric  current  (see  Chapter  IV). 

When  the  two  points  1  and  2  are  an  infinitesimal  distance  dl 
apart,  the  drop  of  potential  along  dl  is  likewise  an  infinitesimal 
quantity  and  may  be  written  —dV  where  dV  stands  for  an  in- 
finitesimal increase  of  potential  along  dl  and  therefore  —  dFfora 
decrease  or  drop  of  potential  along  dl.  We  then  have  that 
-dV  =  (H  cos9)dl 

n        dV 
or  H  cos  u  = 

dl  (25a) 


92  ELECTRICAL  ENGINEERING 

That  is,  the  component  in  any  direction  of  the  field  intensity  at 
any  point  is  equal  to  the  negative  of  the  "  space  rate"  of  change  of  the 
potential  in  that  direction.     When  the  elementary  length  is  taken 
in  the  direction  of  the  field  intensity  equation  (25a)  becomes 
H  =  _dV  (256) 

dlt 

where  dlt  means  an  elementary  length  taken  tangent  to  the  line 
of  force  through  this  point.  Hence  a  large  field  intensity  cor- 
responds to  a  rapid  fall  of  potential  in  the  direction  of  the  field 
intensity,  and  a  small  field  intensity  a  gradual  fall  of  potential  in 
the  direction  of  the  field  intensity.  Consequently,  the  expression 
"  potential  gradient  "  is  frequently  used  for  field  intensity,  where 
by  "  potential  gradient "  is  meant  the  drop  of  potential  per  unit 
length  in  the  direction  of  the  lines  of  force.  The  unit  of  magnetic 
potential  difference  in  the  c.  g.  s.  system  is  called  the  gilbert. 
Hence  magnetic  field  intensity  may  be  expressed  as  so  many 
gilberts  per  centimeter. 

When  the  magnetic  field  is  produced  by  an  electric  current  it 
is  usual  to  express  the  drop  of  magnetic  potential  as  so  many 
"  ampere-turns  "  (see  Chapter  IV.).  The  relation  between  gilberts 
and  ampere-turns  is 

1  gilbert  =0.79578  ampere-turns. 

61.  Equipotential  Surfaces.  —  A  magnetic  equipotential  sur- 
face is  a  surface  drawn  in  a  magnetic  field  in  such  a  mariner  that 
the  drop  of  magnetic  potential  along  any  path  in  the  surface  is 
zero.  Such  a  surface  is  perpendicular  at  each  point  to  the  line  of 
force  through  that  point,  otherwise  the  field  intensity  at  the 
point  in  question  would  have  a  component  along  the  surface  and 
consequently  there  would  be  a  difference  of  potential  between 
this  point  and  the  neighboring  point  in  the  direction  of  this  com- 
ponent. The  lines  of  force  representing  a  magnetic  field  are 
therefore  always  normal  to  any  equipotential  surface  which  may 
be  drawn  in  the  field.  Calling  dn  an  elementary  length  measured 
outward  along  the  normal  to  such  a  surface  at  any  point,  the 
field  intensity  at  this  point  is 

dV  (26) 

H  =  —"3 
an 

which  is  the  same  as  equation  (256),  only  expressed  in  a  differ- 
ent form. 


MAGNETISM  93 

SUMMARY    OF    IMPORTANT    DEFINITIONS 
AND  PRINCIPLES 

1 .  A  unit  point-pole  is  a  pole  which  repels  with  a  force  of  one 
dyne  an  equal  point-pole  placed  one  centimeter  away. 

2.  Two  point-poles  of  strengths  m  and  m'  at  a  distance  r  centi- 
meters apart  repel  each  other  with  a  force  of 

m  m'    . 

f=—  —  dynes. 
r2 

3.  The  field  intensity  at  any  point  of  a  magnetic  field  is  defined 
as  the  force  in  dynes  which  would  act  on  a  unit  north  point-pole 
placed  at  that  point  due  solely  to  the  agents  (magnetic  poles  or 
electric  currents)  producing  the  original  field.     The  unit  of  field 
intensity  in  the  c.  g.  s.  system  is  the  gilbert  per  centimeter. 

4.  The  field  intensity  at     a  distance  r  from  a  point-pole  of 
strength  m  is 

TT     m    -n 

ti  =—  gilberts  per  cm. 

r2 

5.  The  mechanical  force  exerted  on  a  point-pole  of  strength  m 

is 

F=mH  dynes 

where  H  is  the  field  intensity  in  gilberts  per  cm.  at  the  point 
occupied  by  m  due  to  all  the  poles  (and  electric  currents)  in  the 
field  except  the  pole  m. 

6.  The  normal  component  of  the  field  intensity  at  any  point 
P  just  outside  a  magnetically  charged  surface  due  solely  to  the 
pole  on  this  surface  is 

H=2  TT  a  gilberts  per  cm. 

where  cr  is  the  pole  strength  per  unit  area  at  the  point  on  the 
surface  directly  opposite  P. 

7.  The  magnetic  moment  of  a  magnet  is  defined  as  the  ratio 
of  the  maximum  moment  exerted  on  a  magnet,  when  placed  in  a 
uniform  magnetic  field,  to  the  intensity  of  this  field.     The  magnetic 
moment  of  a  bar  magnet  of  length  I  with  poles  of  strength  m 
and  — m  concentrated  in  points  at  its  two  ends  is  ml. 

8.  The  frequency  of  vibration  of  a  magnetic  needle  suspended 
in  a  magnetic  field  is  proportional  to  the  square  root  of  the  field 
intensity. 

9.  Lines  of  magnetic  force  are  lines  drawn  in  a  magnetic  field 
in  such  a  manner  that  they  coincide  in  direction  at  each  point  P 
with  the  field  intensity  at  P  and  their  number  per  unit  area  at  each 


94  ELECTRICAL  ENGINEERING 

point  P  across  a  surface  at  right  angles  to  their  direction  at  P  is 
equal  to  the  field  intensity  at  P. 

10.  The  number  of  lines  of  force,  or  flux  of  force,  crossing  any 
elementary  area  ds  is 

d\l*=(H  cos  a)  ds 

where  H  is  the  field  intensity  at  ds  and  a  is  the  angle  between  the 
direction  of  H  and  the  normal  to  ds;  or  H  cos  a  is  the  component 
of  H  normal  to  ds. 

11.  Gauss's  Theorem: — The  algebraic  sum  of  the  number  of 
lines  of  force  outward  from  any  closed  surface  is  equal  to  4  TT  times 
the  algebraic  sum  of  the  poles  inside  this  surface,  i.  e.} 


(H  cos  a)  ds  =4  TT  2,  m 

\s\ 

where  /     represents  the  surface  integral  over  the  closed  surface  S. 
J  \s\ 

12.  A  magnetised  body  is  considered  to  be  made  up  of  mag- 
netic filaments  such  that  were  the  lateral  walls  of  any  one  of  these 
filaments  separated  from  the  rest  of  the  body  by  a  narrow  air  gap, 
no  poles  would  be  formed  on  these  lateral  walls. 

13.  The  intensity  of  magnetisation  J  at  any  point  of  a  mag- 
netised body  is  the  value  of  the  strength  per  unit  area  of  the  pole 
which  would  be  formed  on  the  walls  of  a  gap  cut  in  the  body  at  this 
point  perpendicular  to  the   direction  of  the   magnetic   filament 
through  this  point.     The  direction  of  the  intensity  of  magnetisa- 
tion is  the  direction  of  the  filament,  and  the  positive  sense  of  the 
filament  is  the  sense  of  the  line  drawn  into  the  gap  from  the  wall  on 
which  the  north  pole  is  formed. 

14.  A  line  of  magnetisation,  or  unit  magnetic  filament,  is  a 
filament  of  such  a  size  that  were  it  broken  by  a  narrow  gap  at  any 
point,  the  strength  of  the  pole  formed  on  either  wall  of  the  gap 

would  be  — . 

15.  The  number  of  lines  of  magnetisation  crossing  any  ele- 
mentary area  ds  is 

d  N  =4  TT  ( Jcos  a)  ds 

where  J  is  the  intensity  of  magnetisation  at  ds  and  a  is  the  angle 
between  the  direction  of  J  and  the  normal  to  ds. 

16.  The  number  of  lines  of  magnetic  induction,  or  flux  of  in- 
duction, crossing  any  surface  is  defined  as  the  algebraic  sum  of 


MAGNETISM  95 

the  lines  of  magnetisation  and  the  lines  of  force  crossing  that 
surface.  The  direction  of  the  line  of  induction  at  any  point  is 
the  direction  of  the  vector  which  is  equal  to  the  vector  sum  of  4  TT  J 
and  H  at  this  point.  The  unit  of  flux  of  induction  in  the  c.  g.  s. 
system  is  the  maxwell;  one  line  of  induction  is  equal  to  one 
maxwell. 

17.  The  flux  density  B  at  any  point  is  denned  as  the  number  of 
lines  of  induction  per  unit  area  crossing  an  elementary  surface 
drawn  at  this  point  normal  to  their  direction.     The  flux  density 
is  the  vector  sum 

5=477  J+H. 

The  unit  of  flux  density  in  the  c.  g.  s.  system  is  the  gauss. 

18.  The  number  of  lines  of  induction  crossing  any  elementary 
area  ds  is 

d  (f)=(B  cos  a)  ds 

where  a  is  the  angle  between  the  direction  of  B  and  the  normal  to 
ds: 

19.  A  line  of  force  originates  at  a  north  pole  and  ends  at  a 
south  pole  and  may  exist  in  either  a  magnetic  or  non-magnetic 
substance.     A  line  of  magnetisation  originates  at  a  south  pole 
and  ends  at  a  north  pole  and  exists  in  a  magnetic  substance  only. 
A  line  of  induction  is  a  closed  loop  without  ends  and  may  exist  in 
either  a  magnetic  or  non-magnetic  substance.     In  a  non-magnetic 
substance  lines  of  force  and  lines  of  induction  are  identical,  in  a 
magnetic  substance  they  are  never  identical. 

20.  The  magnetic  permeability  //,  of  a  body  is  the  ratio  of  the 
flux  density   B  established  in  the  body  to  the  resultant    field 
intensity  H,  when  the  body,  originally  unmagnetised,  is  placed  in 
a  magnetic  field  which  is  continually  increased  from  zero  to  the 
value  H;  that  is 

B 

^~H 
The  flux  density  is  not  a  constant  but  depends  upon  the  value  of  H. 

21.  When  the  field  intensity  in  a  magnetic  body  is  changed 
from  one  value  to  another  and  then  back  again  to  its  original  value, 
the  flux  density  does  not  return  to  its  original  value.     This  phe- 
nomenon is  known  as  magnetic  hysteresis.     As  a  result  of  this 
phenomenon  a  certain  amount  of  heat  energy  is  always  dissipated 
in  a  magnetic  body  when  it  is  subjected  to  a  varying  magnetic  field. 


96  ELECTRICAL  ENGINEERING 

22.  The  magnetic  potential  V  at  any  point  in  a  magnetic  field 
is  the  work  which  would  be  done  by  the  agents  producing  the  field 
in  moving  a  unit  north  point-pole  from  this  point  to  infinity.     The 
unit  of  magnetic  potential  in  the  c.  g.  s.  system  is  the  gilbert.     The 
potential  at  any  point  due  to  a  point-pole  of  strength  m  at  a  dis- 
tance r  centimeters  away  is 

T7-  Wl          MT 

V  =  —  gilberts. 
r 

The  resultant  potential  due  to  any  number  of  poles  is  the  algebraic 
sum  of  the  potentials  due  to  all  the  individual  poles. 

23.  The  drop  of  potential  from  any  point  1  to  any  point  2  is 

C2 
V, 


\-V2  =  f\H  cos  0)  dl gilberts 


where  dl  is  the  length  in  centimeters  of  any  element  of  the  path 
from  1  to  2,  H  the  field  intensity  in  gilberts  per  cm.  at  dl,  and 
0  the  angle  between  the  direction  of  H  and  dl.  When  the  field  is 
due  solely  to  magnetic  poles  the  drop  of  potential  is  independent  of 
the  path  from  1  to  2.  This  is  not  true  when  the  path  links  an 
electric  current. 

24.  A  magnetic  equipotential  surface  is  a  surface  drawn  in  a 
magnetic  field  in  such  manner  that  the  drop  of  magnetic  potential 
along  any  path  in  the  surface  is  zero.  Such  a  surface  is  per- 
pendicular at  each  point  to  the  line  of  force  through  that  point. 

PROBLEMS 

(Note :  In  calculating  forces  and  field  intensities  when  a  slender 
bar  magnet  is  specified,  the  poles  are  to  be  considered  concentrated 
in  points  at  its  ends.) 

1.  Two  slender  bar  magnets  each  with  poles  of  100  c.  g.  s.  units 
and  10  inches  in  length  lie  upon  the  same  straight  line  with  their 
centers  15  inches  apart.     What  is  the  force  in  dynes  exerted  by  one 
magnet  on  the  other  if  the  nearest  poles  on  the  respective  magnets 
are  of  opposite  signs? 

Ans. :  50.7  dynes. 

2.  Two  slender  bar  magnets  A  and  B  are  placed  parallel  to 
each  other  and  10  inches  apart  with  their  centers  on  a  line  per- 
pendicular to  their  axes  and  with  opposite  poles  on  the  respective 
magnets  adjacent.     The  poles  of   A  are  50  c.  g.  s.  units  and  the 
poles  of  B  are  30  c.  g.  s.  units.      A  is  20  inches  in  length  and  B  is  10 


MAGNETISM  97 

inches  in  length.     What  is  the  amount  (in  dynes)  and  the  direction 
of  the  total  force  exerted  by  one  magnet  on  the  other? 

Ans.".  2.52  dynes  in  the  direction  perpendicular  to  the  axis  of 
each  magnet. 

3.  Find  the  amount  and  direction  of  the  field  intensity  due  to 
a  slender  bar  magnet  at  a  point  10  cm.  from  the  magnet  on  a  line 
normal  to  the  axis  of  the  magnet  through  its  north  pole.     The 
magnet  is  10  cm.  in  length  and  the  strength  of  each  pole  is  50  c.  g.  s. 
units. 

Ans.:  0.368  gilberts  per  cm.  at  an  angle  of  61.3°  with  the  axis  of 
the  magnet. 

4.  A  slender  bar  magnet  with  poles  of  25  c.  g.  s.  units  is  placed  in 
a  uniform  field  the  intensity  of  which  is  100  gilberts  per  cm.     What 
is  the  force  in  dynes  acting  upon  each  pole?    What  is  the  total 
force  acting  on  the  magnet? 

Ans.:  2500  dynes  on  each  pole ;  total  force  zero. 

5.  Two  similar  bar  magnets  A  and  B  are  placed  end  on  with 
their  two  nearest  pole-faces  -fa  inch  apart  and  of  opposite  sign. 
Each  magnet  is  20  inches  long  and  the  pole  strength  of  each  pole  is 
100  c.  g.  s.  units  uniformly  distributed  over  the  respective  end  sur- 
faces of  the  magnet.     If  the  cross  section  of  each  magnet  is  1 
square  inch,  find  the  force  in  dynes  exerted  on  each  other  by  the 
two  adjacent  poles.     What  is  the  force  produced  by  one  magnet 
on  the  other  due  to  the  action  of  the  other  poles? 

Ans.:  An  attraction  of  9736  dynes.  A  repulsion  of  6.79  dynes. 
(Note  that  this  latter  force  is  less  than  0.1%  of  the  force  due  to  the 
adjacent  poles.) 

6.  A  slender  bar  magnet  30  cm.  in  length  and  with  poles  of 
80  c.  g.  s.  units  is  pivoted  at  its  center  and  placed  in  a  uniform  mag- 
netic field,  the  intensity  of  which  is  300  gilberts  per  cm.     What  is 
the  torque  acting  upon  the  magnet  about  its  center,  when  the  axis 
of  the  magnet  is  perpendicular  to  the  field  intensity?     What  is 
the  magnetic  moment  of  the  magnet? 

Ans.:  720,000  cm-dynes.     2400  c.  g.  s.  units. 

7.  A  slender  bar  magnet  I  centimeters  long  produces  a  field 
intensity  of   H  gilberts  per  cm.  at  a  point  on  a  line  through  the 
axis  of  the  magnet  at  a  distance  of  101  centimeters  from  its  center. 
What  is  the  field  intensity  produced  by  this  magnet  at  a  point  on  a 
line  through  the  center  of  the  magnet  perpendicular  to  its  axis  and 
at  the  same  distance  I  from  its  center? 


98  ELECTRICAL  ENGINEERING 

In  solving  this  problem  note  that  ( — -\  *  is  negligible  compared 
with  unity. 

TT 

Ans.:  —  gilberts   per.  cm.     Note:   The   ratio   of   these  two 

Zi 

intensities  depends  solely  upon  the  exponent  of  r  in  the  funda- 
mental formula,  equation  (1),  for  the  mutual  action  of  two  poles. 
Gauss  determined  this  ratio  experimentally  to  be  2:1,  which  was 
the  first  accurate  experimental  proof  of  the  inverse  square 
law. 

8.  A  slender  bar  magnet  5  square  centimeters  in  cross  section 
and  30  centimeters  in  length  has  a  pole  strength  of  1500  c.  g.  s. 
units.  Find  the  field  intensity  and  the  flux  density  at  the  center 
of  the  magnet ;  state  the  direction  of  each. 

Ans.:  13.3  gilberts  per  cm.  in  the  direction  from  north  to  south 
pole  of  the  magnet.  3757  gausses  in  the  direction  from  south  to 
north  pole  of  the  magnet. 

y  9.  A  slender  iron  bar  3  sq.  cm.  in  cross  section  and  40  cm. 
in  length  is  placed  in  a  uniform  magnetic  field,  the  intensity  of 
which  before  the  introduction  of  the  bar  is  160  ampere-turns  per 
inch.  After  the  bar  of  iron  is  placed  in  this  field  with  its  axis 
•parallel  to  the  direction  of  the  field  intensity,  a  uniform  intensity 
of  magnetisation  of  1000  c.  g.  s.  units  is  established  in  the  bar  (i.e., 
the  lines  of  magnetisation  are  to  be  assumed  straight  lines  parallel 
to  the  axis  of  the  bar).  Calculate  the  field  intensity  in  gilberts  per 
cm.  and  the  total  flux  of  induction  at  the  center  of  the.  bar;  state 
the  direction  of  each. 

Ans.:  64.2  gilberts  per  cm.  in  the  direction  from  the  south  to 
the  north  pole  of  the  bar.  37,773  maxwells  in  the  same  direction 
as  field  intensity. 

10.  The  field  intensity  at  the  center  of  a  slender  bar  magnet 
100  cm.  long  and  4  sq.  cm.  in  cross  section  is  500  gilberts  per  cm. 
If  the  flux  density  at  the  center  of  the  magnet  is  5000  gausses  and 
is  parallel  to  the  direction  of  the  field  intensity,  what  is  the  strength 
of  each  pole,  assuming  the  lines  of  magnetisation  to  be  straight 
lines?     What  would  be  the  field  intensity  in  the  region  occupied  by 
the  magnet  if  the  magnet  were  removed? 

Ans.:  1432  c.  g.  s.  units.     501.15  gilberts  per  cm.  * 

11.  A  slender  magnet  has  a  magnetic  moment  of  5000  c.  g.  s. 
units.     The  magnet  is  50  cm.  in  length  and  2  sq.  cm.  in  cross 


MAGNETISM  99 

section.     Calculate  the  intensity  of  magnetisation  and  the  flux  of 
magnetisation  at  the  center  of  the  rod. 

Ans.:  50  c.  g.  s.  units.     1257  c.  g.  s.  units. 

4  12.  A  slender  bar  of  iron  50  cm.  long  and  5  sq.  cm.  in  cross 
section  is  placed  in  a  uniform  field  the  intensity  of  which  is  50 
gilberts  per  cm.,  the  axis  of  the  iron  bar  being  parallel  to  the 
direction  of  the  field  intensity.  If  the  permeability  of  the  iron  at 
the  degree  of  saturation  attained  is  300,  find  the  intensity  of  mag- 
netisation and  the  flux  density  at  the  center  of  the  bar. 
Ans.:  862  c.  g.  s.  units.  10,870  gausses. 

13.  Fig.  36,  p.  83,  represents  the  hysteresis  loop  for  a  sample 
of  iron.     Calculate  the  torque  required  to  revolve  a  sample  of 
this  iron  having  a  volume  of  50  cu.  cm.  at  a  uniform  speed  of  300 
revolutions  per  minute  in  a  uniform  magnetic  field  which  has  an  in- 
tensity of  12  gilberts  per  cm.     The  demagnetising  action  of  the 
poles   induced  on   the   iron   and   mechanical   friction   are   to   be 
neglected. 

To  solve  this  problem  equate  the  energy  dissipated  per  second 
in  the  iron  due  to  hysteresis  to  the  mechanical  power  expressed 
in  terms  of  torque  and  angular  speed. 

Ans.:  1770  cm-dynes.  (Note  that  this  torque  is  independent 
of  the  speed,  but  depends  only  upon  the  area  of  the  hysteresis 

loop,  i.e.,  the  torque  is   T = •.     Hence  the  torque  is  propor- 

8  7T2 

tional  to  the  hysteresis  loss  per  cycle  of  variation  of  field  inten- 
sity.    This  is  the  principle  of  Ewing's  hysteresis  tester.) 

14.  Three  equal  north  point-poles  of  25  c.  g.  s.  units  each 
are  placed  at  the  vertices  of  an  equilateral  triangle,  each  side  of 
which  is  5  inches  in  length.     Find  (1)  the  intensity  of  the  field  at 
the  center  of  the  triangle ;  (2)  the  magnetic  potential  at  the  center 
of  the  triangle ;  (3)  the  intensity  of  the  field  at  a  point  on  a  line 
through  any  two  poles  midway  between  them ;  and  (4)  the  magnetic 
potential  at  a  pointvon  a  line   through  any  two  poles  midway 
between  them.     South  poles  may  be  neglected. 

Ans.:  (1)  0.  (2)  26.0  gilberts.  (3)  1.33  gilberts  per  cm. 
in  the  direction  away  from  the  third  pole.  (4)  25.8  gilberts. 

15.  The  numerical  strength  of  each  pole  of  a  round  bar  magnet 
is  200  c.  g.  s.  units,  and  these  poles  are  uniformly  distributed  over 
its  end  surfaces.     The  cross  section  of  the  magnet  is  1  sq.  in.  and 
its  length  is  10  inches.     Calculate  the  magnetic  potential  at  the 


100  ELECTRICAL  ENGINEERING 

center  of  the  north  pole  of  the  magnet.  In  making  this  calculation 
find  first  the  potential  at  this  point  due  to  the  north  pole  con- 
sidered as  a  uniformly  magnetically  charged  disc  and  then  add 
(algebraically)  to  this  the  potential  due  to  the  south  pole  con- 
sidered as  a  point-pole.  Why  is  this  approximation  justified? 
What  is  the  drop  of  magnetic  potential  from  one  pole  to  the  other? 
Is  the  drop  through  the  magnet  the  same  as  the  drop  between  these 
two  points  through  the  air  around  the  magnet? 

Ans.:  1249  gilberts.     Since  the  angle  subtended  by  the  south 
pole  is  small,  see  Article  37.     2498  gilberts.     Yes. 


Ill 


CONTINUOUS  ELECTRIC  CURRENTS 

62.  The  Electric  Current.  —  When  a  strip  of  zinc  is  dipped 
into  a  dilute  solution  of  sulphuric  acid  in  water,  hydrogen  gas 
is  given  off  from  the  strip,  but  if  the  zinc  is  pure,  this  action 
ceases  almost  immediately.  A  copper  strip  dipped  into  the 
same  solution  is  not  appreciably  affected,  provided  the  two  strips 


B 


Copper  Strip- 


Dilute 
Sulphuric  Acid 


.Zinc  Strip 


Fig.  43. 


do  not  touch  each  other.  Now  connect  the  ends  of  the  two  strips 
by  a  copper  wire  (Fig.  43);  the  following  phenomena  are  then 
observed  : 

1.  The  zinc  gradually  wastes  away  and  hydrogen  gas  is  lib- 
erated at  the  copper  strip. 

2.  The  wire,  the  strips,  and  the  solution  all  become  heated. 

3.  A  magnetic  field  is  produced  around  the  wire,  the  strips  and 
the  solution. 

4.  A  magnet  placed  in  the  vicinity  of  the  apparatus  will  exert 
a  mechanical  force  on  the  wire,  the  strips,  and  the  solution. 

5.  When  the  copper  wire  connecting  the  two  strips  is  broken  in 
two  and  the  two  ends  a  and  b  (Fig.  44)  are  both  dipped  into  a 
solution  of  copper  sulphate,  it  is  found  that  the  wire  a  wastes 
away  where  it  is  immersed  in  the  copper  sulphate  solution  and 
copper  is  deposited  on  the  wire  b  where  it  is  immersed  in  the  solu- 

101 


102 


ELECTRICAL  ENGINEERING 


tion.  In  addition,  this  solution  becomes  heated,  has  a  magnetic 
field  produced  around  it,  and  a  magnet  placed  near  it  exerts  a 
mechanical  force  upon  it. 

6.    When  the  ends  A  and  B  of  the  copper  wire  are  reversed,  so 


Copper 
Stri: 


1 

rA 

1 

L 

B 

.Zinc  Strip 

b 

a 

Copper  Sulphate 

Solution  

Fig.  44. 

that  A  is  connected  to  the  zinc  strip  and  B  is  connected  to  the 
copper  strip,  all  the  phenomena  remain  the  same  as  before  except 
that 

a.  The  direction  of  the  magnetic  field  around  the  wire  and  the 
copper  sulphate  solution  is  reversed. 

6.  The  direction  of  the  mechanical  force  produced  on  the  wire 
and  the  copper  sulphate  solution  by  a  magnet  placed  in  their 
vicinity  is  also  reversed. 

c.  The  end  of  the  wire  a  in  the  copper  sulphate  solution  now 
has  copper  deposited  upon  it,  while  the  end  of  the  wire  b  in  the 
copper  sulphate  solution  wastes  away. 

These  various  phenomena  are  said  to  be  due  to  a  "  current 
of  electricity,"  or  briefly,  to  an  "  electric  current  "  in  the  wire,  the 
strips  and  the  solution.  The  various  parts  of  the  apparatus 
taken  together  are  said  to  form  a  closed  electric  circuit,  and  any 
one  of  the  parts  is  spoken  of  as  part  of  an  electric  circuit.  The 
combination  of  the  copper  and  zinc  strips  and  the  dilute  sulphuric 
acid  solution  is  called  an  electric  battery,  and  the  copper  and  zinc 
strips  are  called  the  poles  of  the  battery.  Experiment  shows  that 
when  the  phenomena  just  described  occur  in  and  around  the  wire, 
the  battery  loses  chemical  energy,  consequently  the  battery  is 
to  be  looked  upon  as  the  cause  of  these  phenomena,  and  since  the 
phenomena  are  attributed  to  a  flow  of  "  electricity  "  through  the 
various  parts  of  the  apparatus,  the  battery  is  said  to  produce 
an  "  electricity  moving  force  "  or,  more  briefly,  an  electromotive 
force.  There  are  other  devices  which  are  capable  of  causing 
these  phenomena  to  occur  in  and  around  a  wire  connected  to 


CONTINUOUS  ELECTRIC  CURRENTS  103 

them,  and  any  such  device  is  said  to  be  the  seat  of  an  electro- 
motive force.  The  electric  dynamo,  the  operation  of  which 
will  be  described  later,  is  a  device  which  is  capable  of  producing 
very  powerful  electromotive  forces  and  electric  currents  thou- 
sands of  times  greater  than  the  maximum  current  which  can  be 
produced  by  a  simple  battery  of  the  form  just  described.  But 
before  we  can  speak  accurately  of  the  quantitative  values  of 
electric  currents  and  electromotive  forces,  it  is  necessary  to  define 
what  shall  be  taken  as  the  measure  of  these  quantities. 

63.  Conductors  and  Insulators  or  Dielectrics.  —  First,  however, 
it  is  important  to  note  that  the  degree  to  which  the  effects  described 
above  are  produced  when  different  kinds  of  substances  are  used 
in  place  of  the  copper  wire  to  connect  the  poles  of  the  battery, 
depends  to  a  very  great  extent  upon  the  nature  of  these  substances. 
For  example,  when  there  is  nothing  but  air  between  the  poles  of 
the  battery  none  of  the  effects  described  above  are  observed. 
Again,  a  dry  silk  string  may  be  used  to  connect  the  poles  of  the 
battery,  and  no  effects  will  be  observed.  When  a  moist  string  is 
used,  the  effects  produced  are  similar  to  those  produced  when  the 
poles  are  connected  by  the  copper  wire,  but  to  a  markedly  less 
degree.  Any  substance  which,  when  connected  to  the  poles  of  an 
electric  battery,  has  a  magnetic  field  produced  around  it  as  long  as 
it  remains  in  contact  with  these  poles,  is  called  a  conductor  of 
electricity;  if  no  magnetic  field  is  established  around  the  substance, 
it  is  called  an  insulator  or  dielectric.  It  should  be  noted,  however, 
that  when  a  battery  of  sufficiently  high  electromotive  force  is 
used,  any  substance  connected  to  its  poles  will  have  a  magnetic 
field  produced  around  it,  though  to  a  far  less  extent  than  would 
be  produced  around  a  metallic  wire  connected  to  its  poles.  In 
fact,  there  is  no  known  substance  which  is  a  perfect  insulator, 
though  the  results  of  all  known  experiments  lead  us  to  believe 
that  a  perfect  vacuum,  if  obtainable,  would  be  such.  However, 
for  most  practical  purposes,  such  substances  as  glass,  glazed  por- 
celain, rubber,  ebonite,  gutta-percha,  paraffine,  silk,  cellulose  and 
shellac  may  be  considered  as  insulators,  while  all  metals,  carbon, 
fused  salts  and  solutions  of  most  mineral  salts  and  acids  are 
conductors. 

A  conductor  completely  surrounded  by  insulators  is  said  to  be 
completely  insulated.  A  wire  is  also  said  to  be  insulated  when  its 
lateral  walls  only  are  surrounded  by  an  insulator.  That  is,  a 


104  ELECTRICAL   ENGINEERING 

rubber-covered  wire,  for  example,  with  its  ends  connected  to  the 
poles  of  a  battery,  is  spoken  of  as  an  "  insulated  wire  connected 
to  the  battery." 

64.  Electricity  Analogous  to  an  Incompressible  Fluid  Filling 
all  Space.  —  The  following  analogies  will  be  found  helpful  in 
understanding  the  significance  of  the  various  properties  which 
experiment  leads  us  to  assign  to  the  something  called  electricity. 
In  the  first  place,  experiment  shows  that  this  something  must 

Elastic     nave  man7  properties  anal- 
'waiu    ogous  to  those  which  would 
be  possessed   by  an  incom- 
pressible fluid  filling  all  space, 
including  the  space  occupied 
by  matter.     In  this  analogy, 
the  force  of  gravitation  act- 
ing on  the  fluid  is  to  be  neg- 
lected.    The  properties  pos- 
45-  sessed  by  free  space  or  any 

space  occupied  by  a  dielectric  are  found  to  be  analogous  to  the 
properties  which  would  be  possessed  by  this  space  were  the 
incompressible  fluid  in  this  space  enclosed  in  minute  cells  with 
elastic  walls,  forming  a  cellular  or  honey-comb  structure  with 
continuous  walls  completely  filling  this  space;  while  the  prop- 
erties possessed  by  a  conductor  are  analogous  to  the  properties 
possessed  by  a  space  in  which  the  walls  of  the  cells  are  completely 
destroyed.  Any  particle  of  this  fluid  in  the  space  corresponding 
to  a  dielectric  can  then  move  only  as  the  result  of  a  strain  pro- 
duced in  the  elastic  structure  which  enmeshes  it,  while  any  particle 
of  the  fluid  in  the  space  corresponding  to  a  conductor  can  move 
freely  throughout  this  space.  A  conductor  may  then  be  looked 
upon  as  analogous  to  an  elastic  sack  (or  tube,  in  the  case  of  a  wire) 
completely  filled  with  this  incompressible  fluid,  the  elastic  walls 
of  the  sack  being  formed  by  the  walls  of  the  cellular  structure 
representing  the  dielectric  surrounding  the  conductor. 

The  property  possessed  by  a  battery  of  being  capable  of  pro- 
ducing a  "  flow  of  electricity  "  is  analogous  to  the  pressure  pro- 
duced by  a  pump.  The  analogy  to  a  simple  battery  of  the  form 
described  above  is  a  pump  which  maintains  a  constant  pressure 
(approximately)  in  the  same  direction  whether  or  not  there  is  a 
flow  of  the  fluid  through  the  pump.  Since  a  battery  is  always 


CONTINUOUS  ELECTRIC  CURRENTS  105 

made  of   conductors,  the  walls  of  this  pump  must  also  be  con- 
sidered as  elastic. 

A  conductor  connected  to  the  two  poles  of  a  battery  is  then 
analogous  to  the  rubber  hose  completely  filled  with  water  (which 
may  be  taken  as  an  approximately  incompressible  fluid)  with 
its  two  ends  connected  respectively  to  the  outlet  and  intake  of  the 
pump,  which  is  also  completely  filled  with  water.  The  pump  will 
then  force  a  current  of  water  through  the  hose,  and  the  strength 


.Constant  Pressure  Pump 


Cellular  Structure 
with  Elastic  Walls 


Rubber  Hose 

Fig.  46. 

of  this  water  current  (i.e.,  the  quantity  of  water  per  unit  time  flow- 
ing through  any  cross  section  of  the  hose)  will  depend  upon  the 
pressure  developed  by  the  pump  and  the  resistance  due  to  the 
friction  of  the  water  against  the  walls  of  the  hose.  The  effects 
produced  in  and  around  the  wire  connected  to  the  battery  are 
similarly  found  to  depend  upon  a  property  of  the  battery  (called 
its  electromotive  force)  analogous  to  the  pressure  developed  by 
the  pump,  and  upon  a  property  of  the  wire  (called  its  resistance) 
analogous  in  certain  respects  to  the  resistance  of  the  rubber  hose 
to  a  water  current. 

When  the  current  of  water  is  established  in  the  hose,  a  fall  of 
pressure  is  produced  in  the  hose  in  the  direction  of  the  flow  of  the 
water,  and  consequently  the  pressure  acting  on  the  walls  of 
the  hose  will  vary  from  point  to  point.  Hence,  near  the  outlet  of 
the  pump  the  walls  of  the  hose  will  expand  and  near  the  intake 
the  walls  will  contract.  Consequently,  there  will  be  more  water  in 
the  portion  of  the  hose  near  the  outlet  of  the  pump  an<J  less  near 
the  intake,  than  there  was  in  these  portions  of  the  hose  before  it 
was  connected  to  the  pump.  Experiment  shows  that  an  analogous 


106  ELECTRICAL  ENGINEERING 

phenomenon  occurs  when  a  wire  is  connected  to  an  electric  bat- 
tery. The  portions  of  the  wire  near  the  two  poles  of  the  battery 
manifest  properties  which  they  did  not  possess  before;  these  prop- 
erties are  such  that  they  may  be  attributed  to  an  excess  of 
electricity,  or  a  "  positive  charge  "  of  electricity  on  the  portion 
of  the  wire  nearer  the  pole  of  the  battery  from  which  the  electric 
current  is  said  to  flow,  and  a  deficit  of  electricity,  or  a  "  negative 
charge  "  of  electricity,  on  the  portion  of  the  conductor  nearer  the 
pole  of  the  battery  toward  which  the  current  is  said  to  flow. 

On  account  of  the  inertia  of  a  mass  of  water,  time  is  required  to 
set  the  water  in  the  hose  in  motion  when  the  latter  is  connected  to 
the  pump.  It  is  also  found  that  time  is  required  for  the  effects 
which  we  ascribe  to  the  flow  of  an  electric  current  to  reach  a  steady 
state.  As  we  shall  see  later,  the  property  of  electricity  which  is 
analogous  to  inertia  may  be  expressed  in  terms  of  the  magnetic 
field  produced  by  electricity  in  motion.  There  is  no  experimental 
evidence  to  lead  us  to  assign  to  electricity  a  property  analogous  to 
weight. 

The  expansion  of  the  walls  of  the  rubber  hose  by  the  pressure 
produced  by  the  pump  likewise  produces  a  pressure  on  the  cellular 
structure  representing  the  dielectric  surrounding  the  wire,  and 
produces  a  displacement  of  the  cells  and  the  fluid  (e.  g.,  water) 
which  they  contain.  When  the  current  of  water  in  the  hose  be- 
comes steady,  the  motion  of  these  cells  ceases,  but  while  the  cur- 
rent in  the  hose  is  being  established,  there  is  also  a  motion  of  these 
cells  and  the  water  they  contain.  As  we  shall  see  later,  an  effect 
is  produced  in  the  dielectric  while  the  current  in  the  wire  is  being 
established  or  changes  in  any  way,  which  may  be  attributed  to 
a  flow  or  displacement  of  electricity  through  the  dielectric,  anal- 
ogous to  the  displacement  of  the  water  in  the  cellular  structure 
surrounding  the  rubber  hose,  which  displacement  ceases  when  the 
electric  current  reaches  a  constant  value. 

The  displacement  of  the  cells  of  the  structure  surrounding  the 
rubber  hose  produces  a  change  in  the  shape  of  these  cells,  and 
therefore  a  strain  in  their  walls.  Analogous  effects  are  observed 
in  the  dielectric  surrounding  a  wire  connected  to  a  battery  or 
other  device  which  is  capable  of  producing  the  phenomena  which 
we  attribute  to  the  flow  of  an  electric  current.  This  effect  is  par- 
ticularly noticeable  when  the  "  electromotive  force  "  of  the  cur- 
rent-producing device  is  high,  and  may  even  cause  a  rupture  of 


CONTINUOUS  ELECTRIC  CURRENTS  107 

the  dielectric,  i.e.,  an  electric  spark.  A  similar  effect  would  be 
produced  should  the  pressure  developed  by  the  pump  produce  a 
strain  in  the  walls  of  the  cells  in  the  structure  around  the  hose 
sufficient  to  break  the  walls  of  these  cells. 

When  the  outlet  and  intake  of  the  pump  which  we  have  been 
considering  are  closed,  we  have  an  arrangement  analogous  to  an 
electric  battery  with  its  poles  insulated  from  each  other.  Since 
our  pump  is  assumed  to  maintain  a  constant  pressure  independent 
of  the  current  flowing  through  it,  and  since  the  walls  of  the  pump 
are  elastic,  near  the  outlet  these  walls  will  be  stretched  and  near 
the  intake  contracted.  Consequently  there  will  be  an  excess  of 
water  near  the  outlet  and  a  deficit  near  the  intake,  compared  to  the 
quantity  of  water  that  would  be  in  these  portions  were  there  no 
pressure  produced  by  the  pump.  The  two  poles  of  a  battery  pos- 
sess analogous  properties,  which  may  be  attributed  to  an  excess 
of  electricity,  or  "  positive  charge  "  of  electricity,  on  the  pole  of 
the  battery  from  which  the  current  is  said  to  flow,  and  a  deficit  of 
electricity,  or  a  "  negative  charge  "  of  electricity,  on  the  pole 
toward  which  the  current  is  said  to  flow.  The  cellular  structure 
surrounding  the  pump  will  likewise  be  in  a  state  of  strain,  due  to 
the  pressure  produced  in  it  by  the  pump.  Experiment  shows  that 
analogous  effects  are  also  produced  in  the  dielectric  surrounding 
the  battery. 

Again,  if  a  closed  sack  which  has  elastic  walls  is  connected  to 
the  outlet  of  the  pump,  more  water  will  be  forced  into  the  sack  by 
the  pump;  while  if  this  sack  is  connected  to  the  intake  of  the  pump 
water  will  be  drawn  out  of  it.  These  effects  are  analogous  to  the 
effects  produced  when  a  conductor  is  connected  to  one  of  the  poles 
of  a  battery  but  not  to  the  other.  When  the  conductor  is  con- 
nected to  one  pole  it  manifests  a  new  property  which  may  be  at- 
tributed to  an  increase  of  electricity  on  it,  or  to  a  "  positive  charge  " 
of  electricity  gained  by  it,  and  when  connected  to  the  other  pole  it 
manifests  a  new  property  which  may  be  attributed  to  a  drain  of 
electricity  from  it,  or  to  a  "  negative  charge  "  of  electricity  on  it. 

It  should  be  noted  that  the  above  discussion  is  merely  a  state- 
ment of  analogies  and  does  not  explain  anything.  These  analogies 
are  useful  as  they  enable  one  to  form  a  picture  of  the  way  the  ob- 
served effects  might  take  place,  but  the  exact  mechanism  of  these 
effects  may  be  entirely  different.  Like  all  other  analogies,  the 
above  must  not  be  pressed  too  far.  For  example,  the  elastic  hose 


108  ELECTRICAL  ENGINEERING 

or  sack  which  we  have  considered  as  representing  a  conductor, 
changes  in  shape  when  connected  to  the  pump,  but  there  is  no  evi- 
dence that  a  wire  or  other  conductor  changes  in  shape  when  it 
manifests  the  properties  which  are  attributed  to  a  "  charge  of 
electricity  "  on  it. 

65.  A  Wire  as  a  Geometrical  Line.  —  In  the  discussion  of  the 
phenomena  which  are  attributed  to  the  flow  of  an  electric  current 
a  wire  will  usually  be  considered  as  equivalent  to  a  geometrical 
line.     This,  of  course,  is  not  strictly  accurate,  since  a  wire  always 
has  a  finite  cross  section.     However,  in  many  cases  the  error 
involved  is  practically  inappreciable;  when  this  is  not  so,  atten- 
tion will  be  called  to  the  fact.     The  exact  expression  for  a  wire 
of   finite   cross   section  may   always   be   derived   when  we  have 
deduced  the  relation  which  holds  for  a  geometrical  line,  for  we 
may  express  this  same  relation  for  the  wire  of  finite  cross  section 
by  considering  the  wire  made  up  of  an  infinite  number  of  fila- 
ments of  infinitesimal  cross  section,  each  of  which  filaments  is 
equivalent  to  a  line,  and  then  determine  the  resultant  effect  due 
to  all  these  filaments.     In  general,  such  an  expression  is  extremely 
difficult  to  evaluate;  only  in  one  or  two  simple  cases  will  it  be 
necessary  to  do  this. 

66.  Definition  of  the  Strength  of  an  Electric  Current. —  Defini- 
tion of  a  Continuous  Current.  —  The  quantity  of  electricity  that 
flows  through  any  section  of  a  wire  in  unit  time  may  be  called 
the  strength  of  the  current  of  electricity  in  this  section  of  the 
wire,  just  as  the  quantity  of  water  flowing  through  any  section 
of  a  pipe  in  unit  time  may  be  called  the  strength  of  the  water 
current  in  this  section  of  the  pipe.     By  "  quantity  "  of  water 
flowing  through  any  section  of  a  pipe  is  meant  the  volume  of 
water  flowing  through  this  section;  therefore  quantity  of  water 
has  a  perfectly  definite  meaning  and  can  be  readily  determined, 
either  directly  or  by  measuring  its  mass.     There  is  no  experi- 
mental evidence,  however,  to  lead  us  to  attribute  to  electricity 
either  mass  or  volume,  in  the  ordinary  sense  of  these  terms.     To 
define  the  strength  of  an  electric  current  in  this  manner,  there- 
fore, it  would  be  necessary  first  to  define  what  is  to  be  meant  by 
"  quantity  "  of  electricity.     It  is  more  convenient,  however,  to 
define  the  strength  of   an  electric   current  in  some  other  way, 
and  then  to  define  "  quantity  "  of  electricity  in  terms  of  the 
electric  current,  particularly  as  this  method  of  procedure  is  in 


CONTINUOUS   ELECTRIC  CURRENTS  109 

accord  with  the  usual  experimental  methods  employed  in  engineer- 
ing work  in  the  determination  of  the  "  quantity  "  of  electricity. 

We  might  take  any  one  of  the  effects  described  above  as  the 
measure  of  the  strength  of  the  current  flowing  in  the  wire.  Scien- 
tists have  agreed,  however,  to  take  as  the  measure  of  the  strength 
of  an  electric  current  flowing  in  a  /' 

wire,  the  mechanical  force  which  / 

is  exerted  on  the  wire  when  it  is  S 

placed  in  a  magnetic  field.  (This 
effect  was  illustrated  above  by 
the  mechanical  force  produced  on 
the  wire  by  a  magnet  placed  near 
it.)  When  a  constant  magnetic  /  Fig.  47. 

field  is  established  at  the  wire  by  some  external  agent  (for  ex- 
ample, by  means  of  a  permanent  magnet)  it  is  found  that  in  gen- 
eral the  force  exerted  on  the  wire  when  it  is  connected  to  a  bat- 
tery of  the  kind  described  in  Article  62  remains  constant  (at 
least  appreciably  so  for  several  seconds  or  more,  though,  unless 
special  precautions  are  taken,  this  force  will  gradually  change). 
The  current  in  the  wire  is  said  to  be  continuous*  as  long  as  this 
force  remains  constant.  This  force  is  found  to  depend  upon 
the  flux  density  at  the  wire  of  the  magnetic  field  produced  by  this 
external  agent  and  also  upon  the  direction  of  this  flux  density 
with  respect  to  the  direction  of  the  wire,  and  upon  the  length  of 
the  wire.  For  a  continuous  current  in  the  wire,  the  force  dF 
produced  on  any  elementary  length  dl  of  the  wire  (see  Fig.  47) 
by  a  magnetic  field  when  the  flux  density  at  dl  is  B  and  makes 
an  angle  9  with  the  direction  of  dl,  is  found  to  be  proportional 
to  the  product  of  the  flux  density  B,  the  sine  of  the  angle  0,  and 
the  length  dl;  that  is,  the  mechanical  force  dF  is  proportional 
to  ( B  sin  9 )  dl.  The  direction  of  this  force  is  found  by  experiment 
to  be  perpendicular  to  the  plane  determined  by  the  direction  of  the 
flux  density  B  and  the  direction  of  the  length  dl. 

Various  modifications  may  be  made  in  the  rest  of  the  circuit 
which  will  cause  the  force  acting  on  a  given  length  of  wire  to 
change,  even  though  the  flux  density  B  and  the  angle  9  at  each 
point  is  kept  unaltered.  For  example,  another  piece  of  wire  may 
be  inserted  between  one  end  of  the  original  wire  and  the  pole  of 

*The  following  definitions  are  those  adopted  by  the  American  Institute 
of  Electrical  Engineers. 


110  ELECTRICAL   ENGINEERING 

the  battery  to  which  it  was  originally  connected,  that  is,  placed 
"  in  series  "  with  the  original  wire;  or  a  second  battery  may  be 
connected  in  the  circuit.  Any  modification  which  causes  a  change 
in  the  force  acting  on  a  given  length  of  wire  when  the  flux  density 
B  and  the  angle  9  at  each  point  is  kept  constant,  is  attributed 
to  a  change  in  the  strength  of  the  electric  current  in  the  wire.  In 
short,  the  strength  of  the  current  in  any  section  of  the  wire  is 
arbitrarily  defined  to  be  proportional  to  this  force  when  the  flux 
density  B  and  the  angle  9  at  each  element  of  this  section  is  kept 
constant.  We  then  have  that  the  force  on  any  elementary  length 
dl,  besides  being  proportional  to  (B  sin  9}  dl  is  also  proportional 
to  the  strength  of  the  current;  that  is 

dF=kI(Bsin0)dl 

where  7  is  the  strength  of  the  current  in  the  elementary  length  dl 
of  the  wire,  B  is  the  flux  density  of  the  magnetic  field  at  dl  pro- 
duced by  any  external  agent,  9  the  angle  between  the  direction 
of  this  flux  density  and  the  direction  of  dl,  and  k  is  a  factor 
of  proportionality.  Experiment  shows  that  this  quantity  k  is 
independent  of  the  nature  of  the  substances  which  form  the  wire 
and  the  medium  surrounding  the  wrre,  but  depends  only  upon 
the  units  in  which  the  force,  the  flux  density,  and  the  length  are 
measured.  The  unit  of  current  strength  may  therefore  be  chosen 
such  that  when  the  force  dF  is  measured  in  dynes,  the  flux  den- 
sity B  .in  gausses,  and  the  length  dl  in  centimeters,  this  factor 
of  proportionality  k  is  equal  to  unity.  The  above  equation  then 
becomes* 

dF=I  (B  sin  9)  dl 
from  which 

dF  (1) 

I  =  (B  sin  9}  dl 

Note  that   —  is  the  mechanical  force  per  unit  length  of  wire 
dl 

at  dl,  and  that  B  sin  9  is  the  'component  of  the  flux  density  at 
dl  perpendicular  to  the  wire  at  this  point.  Hence,  as  the  measure 
of  the  strength  of  the  electric  current  in  a  wire  is  taken  the  ratio 
of  the  force  per  unit  length  of  the  wire  which  would  be  produced 
by  a  magnetic  field  to  the  component  of  the  flux  density  of  this  field 
perpendicular  to  the  wire.  This  definition  and  its  mathematical 
'*This  relation  is  known  as  Biot  and  Savart's  Law. 


CONTINUOUS   ELECTRIC  CURRENTS  111 

expression,  equation  (1),  applies  to  a  variable  as  well  as  a  con- 
tinuous current. 

When  the  force  is  expressed  in  dynes,  the  length  in  centimeters, 
and  the  flux  density  in  gausses,  the  unit  of  electric  current  strength 
as  thus  denned  is  called  the  c.  g.  s.  electromagnetic  unit  of  current, 
or  the  absolute  unit  of  current,  or  the  abampere.  One  abampere 
is  then  equal  to  one  dyne  per  centimeter  per  gauss.  In  practice, 
a  unit  of  one-tenth  the  size  of  this  unit  is  employed ;  this  practical 
unit  is  called  the  ampere.  Hence 

1  abampere  =  10  amperes. 

Instead  of  employing  the  expression  "  An  electric  current  has 
a  strength  of  so  many  amperes  or  abamperes  "  one  usually  says  a 
current  is  so  many  amperes  or  abamperes. 

When  the  element  dl  is  in  a  non-magnetic  medium,  which  is  prac- 
tically always  the  case  in  any  current-measuring  instrument,  the 
flux  density  at  dl  is  equal  to  the  field  intensity  at  this  element, 
and  consequently  in  this  case  equation  (1)  becomes 

/=        dF  (la) 

(H  sinQ)  dl 

where   H  is  the  field  intensity  in  gilberts  per  centimeter  at  the 
element  dl. 

67.  Definition  of  the  Direction  of  an  Electric  Current.  —  Left- 
Hand  Rule.  —  As  noted  in  Article  62,  when  the  ends  of  the  wire 
connected  to  the  poles  of  the  battery  are  interchanged,  the  force 
produced  by  any  external  magnetic  field  on  the  wire  also  reverses. 
An  electric  current  must  therefore  be  looked  upon  as  having 
direction  as  well  as  magnitude.  We  also  saw  in  the  last  paragraph 
that  the  direction  of  the  mechanical  force  on  each  elementary 
length  of  the  wire,  i.e.,  the  direction  in  which  this  elementary 
length  tends  to  move,  is  perpendicular  to  the  plane  determined 
by  this  elementary  length  and  the  direction  of  the  flux  density 
of  the  magnetic  field  at  this  elementary  length.  As  the  direction 
of  the  electric  current  (I)  is  taken  arbitrarily  the  direction  in 
which  the  middle  finger  of  the  left  hand  points  when  the  thumb, 
forefinger  and  middle  finger  of  this  hand  are  held  mutually  per-* 
pendicular,  and  the  thumb  is  pointed  in  the  direction  in  which 
the  wire  tends  to  move  and  the  forefinger  is  pointed  in  the  direction 
of  the  component  of  the  flux  density  perpendicular  to  the  wire. 
This  rule  is  called  the  left-hand  rule;  it  is  readily  remembered 


112 


ELECTRICAL   ENGINEERING 


by  noting  the  corresponding  letters  in  middle  and  I,  thumb  and 
move,  forefinger  and  flux. 

68.  Conductors  in  Series  and  in  Parallel.  —  Experiment  shows 
that  when  a  number  of  conductors  are  connected  end  to  end 
(Fig.  48)  and  are  completely  surrounded  by  insulators,  then  the 
strength  of  the  current  as  above  defined  is  the  same  in  all  these 
conductors,  provided  the  current  strength  does  not  vary  with 

Battery 


Fig.  48. 

time.     Two  or  more  conductors  thus  connected  end  to  end  are 
said  to  be  connected   in  series. 

Experiment  also  shows  that  when  any  portion  of  an  electric 
circuit  between  any  two  points  A  and  B  is  formed  by  two  or 
more  insulated  conductors  (Fig.  49),  the  strength  of  the  current 
coming  up  to  the  junction  point  A  or  leaving  the  junction  point 
B  is  equal  to  the  sum  of  the  strengths  of  the  currents  in  the  con- 


B 


L 


Fig.  49. 

ductors  joining   A   and   B,  provided  the  current  strength  does 
not  vary  with  time.     For  example,  in  Fig.  49 

and  the   conductors   joining   the   two  points  A  and  B  are  said 
to  be  connected  in  parallel. 

When  any  portion  of  a  circuit  is  made  up  of  one  or  more  con- 
ductors connected  in  series  with  one  or  more  groups  of  conductors 
in  parallel,  this  portion  of  the  circuit  is  said  to  be  connected  in 
series-parallel. 


CONTINUOUS  ELECTRIC  CURRENTS  113 

69.  Total  Force  Produced  by  a  Magnetic  Field  on  a  Wire  Carry- 
ing an  Electric  Current.  —  From  the  above  discussion  and  equa- 
tion (1),  it  follows  that  the  total  mechanical  force  F  produced  by 
any  external  magnetic  field  on  an  insulated  wire  carrying  an 
electric  current  7,  when  the  wire  is  of  any  length  I  and  bent  into 
any  shape  whatever,  is  equal  to  the  vector  sum 


(B  sin0)dl  (2) 

where  dl  is  any  elementary  length  of  the  wire  measured  in  the 
direction  of  the  current,  B  the  flux  density  of  the  field  at  dl,  and 

6  the  angle  between  dl  and  the  direction  of   B  and   f   indicates 

the  vector  sum  of  the  expressions  (  B  sin  6}  dl  for  all  the  elementary 
lengths  into  which  the  wire  is  divided.  All  quantities  in  this 
equation  are  in  c.  g.  s.  units.  In  general,  the  flux  density  and 
the  angle  0  will  be  different  for  each  point  of  the  wire.  The 
direction  of  the  mechanical  force  dF  acting  on  each  element  will 
also  be  different  for  each  element,  since  the  plane  determined  by 
the  direction  of  the  flux  density  and  the  direction  of  the  element 
will  in  general  be  different;  hence  the  necessity  for  taking  the 
vector  sum. 

70.  Force  Produced  by  a  Uniform  Magnetic  Field  on  a  Straight 
Wire  Carrying  an  Electric  Current.  —  One  of  the  simplest  cases 
is  that  of  a  straight  wire  in  a  uniform  magnetic  field.     Consider 
such  a  wire  carrying  a  current  7,  placed  at  an  angle  6  with  the 
direction  of  the  lines  of  induction;   let  the  flux  density  of  this 
magnetic  field  be  B.      In  this  case  B  and  6  are  constant  for  all 
points  along  the  wire,  and  the  mechanical  force  on  all  elements 
of  the  wire  is  in  the  same  direction;  the  integration   is  then  a 
simple  algebraic  one  and  therefore  the  total  force  on  the  wire  is 

rl 

F  =IB  sin  0  J  Odl  =IB  I  sin  0.         (2a) 

When  the  wire  is  perpendicular  to  the  direction  of  the  field  the 
force  acting  on  the  wire  is 

F  =  IBl  (2b) 

since  0  =90°  and  sin  90°  =  1.  In  case  the  wire  is  in  a  non-magnetic 
medium,  equation  (26)  becomes 

F=IHl  (2c) 

All  quantities  in  these  equations  are  in  c.  g.  s.  units. 

71.  Magnetic   Field   Produced   by  an   Electric   Current   in  a 
Wire.  —  We  have  seen  that  when  a  wire  carrying  an  electric  cur- 


114  ELECTRICAL  ENGINEERING 

rent  is  placed  in  a  magnetic  field,  this  field  exerts  a  mechanical 
force  on  the  wire.  It  is  also  found  by  experiment  that  an  equal 
and  opposite  mechanical  force  is  exerted  on  the  magnet  or  other 
agent  producing  this  field;  this,  of  course,  is  in  accord  with  the 
general  principle  of  nature  that  "  action  and  reaction  are  equal 
and  opposite."  The  region  around  a  wire  carrying  an  electric 
current  is  therefore  a  magnetic  field  of  force,  for  by  definition 
a  magnetic  field  of  force  is  any  region  in  which  a  magnetic  pole 
will  be  acted  upon  by  a  mechanical  force.  The  intensity  of  the 
field  of  force  due  to  a  wire  carrying  an  electric  current  may  be 
readily  determined  from  equation  (1). 

Consider  an  elementary  length  dl  of  the  wire  in  which  the  cur- 
rent is  7  abamperes  and  let  this  length  be  measured  in  the  direction 

of  the  current. 
Let  the  magnet- 
ic fi  e  1  d  w  h  i  ch 
produces  the 
mechanical  force 
dF  on  this  ele- 
mentary length 
be  that  due  to  a 
unit  north  point- 
'  pole  at  any  point 

P  at  a  distance 
of  r  centimeters  away.     The  flux  density  of  the  magnetic  field  at  dl 

due  to  this  unit  pole  is  then  —  independent  of  the  permeability  of  the 

medium  surrounding  the  pole  and  the  wire  (see  Article  56).  The 
angle  9  between  the  direction  of  this  flux  density  and  the  length 
dl  is  the  angle  between  the  direction  of  dl  and  the  direction  of 
the  line  drawn  from  the  pole  to  dl.  The  mechanical  force  exerted 
by  the  unit  point-pole  on  the  length  dl  is  then 

'sin  0> 


and  is  perpendicular  to  the  plane  determined  by  the  point  P  and 
the  length  dl.  The  direction  of  this  force  is  determined  by  the 
left-hand  rule,  see  Article  67;  in  the  diagram  this  force  is  down 
into  the  plane  of  the  paper.  The  current  /  in  the  elementary 
length  dl  exerts  an  equal  and  opposite  force  on  the  unit  point- 


CONTINUOUS  ELECTRIC  CURRENTS 


115 


pole  at  P.     Therefore  the  intensity  d77t-  of  the  magnetic  field  at 
P  due  to  the  current  7  in  the  elementary  length  dl  is 

_  (7  sin  0}  dl  (3) 


independent  of  the  permeability  of  the  surrounding  medium,  and  is 
in  the  opposite  direction  to  that  of  the  force  on  dl  due  to  the 
pole  at  P.  All  quantities  in  this  equation  are  in  c.  g.  s.  units. 
A  current  /  flowing  in  an  elementary  length  dl  therefore  produces 
a  field  intensity  at  any  point  P  equal  numerically  to  that  pro- 
duced by  a  point-pole  of  strength  (7  sin  0)  dl  coinciding  with  dl, 
where  9  is  the  angle  between  the  direction  of  the  current  in  dl 
and  the  line  drawn  from  P  to  dl,  except  that  this  intensity  does  not 
depend  upon  the  permeability  of  the  surrounding  medium,  whereas 
the  resultant  intensity  at  P  due  to  a  point-pole  at  dl  is  inversely 
proportional  to  the  permeability  of  the  medium  surrounding  P 
and  dl;  see  Article  56.  The  direction  of  this  intensity  is  also 
different  from  that  of  the  intensity  due  to  a  pole  at  dl.  The  latter 
is  in  the  direction  from  dl  to  P,  while  the  intensity  of  the  magnetic 
field  due  to  a  current  in  dl  is  perpendicular  to  the  plane  determined 
by  dl  and  P.  Equation  (3)  applies  to  a  variable  current  as  well' 
as  to  a  continuous  current.  (In  the  case  of  a  rapidly  varying 
current,  however,  the  value  of  H  at  any  instant  does  not  cor- 
respond to  the  value  of  7  at  that  instant,  but  to  the  value  of  7  at 
some  previous  instant,  since  time  is  required  for  the  magnetic 
field  to  be  propagated  through  space;  in  free  space  the  velocity 
of  propagation  is  very  great,  being  the  same  as  the  velocity  of 
light.) 

72.  Direction  of  the  Lines  of  Force  Produced  by  an  Electric 
Current. —  The  lines  of  force  due  to  the  current  7  in  the  elementary 
length  dl  are  circles 
which  have  their 
planes  perpendicular 
to  the  straight  line 
(OO'  in  the  figure) 
drawn  through  dl  and 
which  have  their  cen- 
ters on  this  line.  For, 

the   circumference   of  Fig.  si. 

such  a  circle  is  perpendicular  at  each  point  to  a  plane  drawn 
through  this   point   and   dl;  this   circumference  must   therefore 


O 4-4-H-f- 


dl 


"— O 


116 


ELECTRICAL  ENGINEERING 


coincide  in  direction  with  the  field  intensity  at  this  point;  see 
the  preceding  article.  From  the  deductions  of  the  preceding 
article  it  also  follows  that  the  positive  sense  of  these  lines  of  force 
is  the  same  as  that  in  which  a  right-handed  screw  placed  along 
the  wire  at  dl  must  be  turned  to  advance  it  in  the  positive 
sense  of  the  current.  When  the  positive  senses  of  two  quantities 
are  thus  related,  the  quantities  are  said  to  be  in  the  right- 
handed  screw  direction  with  respect  to  each  other. 

The  current  in  every  other  elementary  length  of  the  circuit 
will  likewise  produce  a  magnetic  field,  the  lines  of  force  of  which 
are  circles  which  have  a  like  relation  to  the  current  in  the  elemen- 
tary length  which  produces  them.  The  lines  of  force  represent- 
ing the  resultant  field  due  to  a  current  in  any  finite  length  of  wire 
will  not  in  general  be  circles,  but  it  can  be  shown  that  each  of  the 
resultant  lines  of  force  is  a  closed  loop  which  "  links  "  the  wire  in 
the  right-handed  screw  direction  with  respect  to  the  current. 

Note  an  important  difference  between  the  lines  of  force  due  to 
an  electric  current  and  those  due  to  magnetic  poles :  the  lines  of 
force  due  to  magnetic  poles  are  not  closed  but  end  on  the  poles,  while 
A  the    lines  of  force  due  to  an  electric 

current  are  always  closed  loops,  link- 
ing the  circuit  which  produces  them. 
When  there  is  any  magnetic  body  in 
the  vicinity  of  an  electric  current  this 
body  will  in  general  have  poles  in- 
duced on  it  by  the  magnetic  field 
due  to  the  current,  and  part  of  the 
lines  of  force  representing  the  result- 
ant field  will  end  on  these  poles  and 
the  rest  will  be  closed  loops  linking 
the  electric  circuit.  The  lines  of 
induction  are  always  closed  loops 
whether  or  not  there  are  magnetic 
poles  in  the  field. 

73.  Magnetic  Field  Due  to  a  Cur- 
rent in  a  Long  Straight  Wire. — A 
useful  application  of  equation  (3)  is 
the  calculation  of  the  intensity  of 
the  magnetic  field  at  any  point  due  to  a  current  in  a  long 
straight  wire. 


dl 


O 


a2\ 


-'P 


B 


Fie.  52. 


CONTINUOUS   ELECTRIC  CURRENTS  117 

In  Fig.  52  let  B  A  be  a  straight  wire  and  let  P  be  any  point  at 
a  distance  r  from  it  (measured  perpendicular  to  the  wire),  I  the 
current  in  the  wire  from  B  to  A  in  abamperes,  dl  any  elementary 
length  in  the  direction  of  /  at  a  distance  I  from  the  point  0  where 
the  perpendicular  from  P  cuts  the  wire,  0  the  angle  between  dl 
and  the  line  from  P  to  dl,  and  x  the  distance  from  P  to  dl.  Each 
element  of  the  wire  will  produce  a  field  intensity  at  P  perpendic- 
ular to  the  plane  of  the  paper,  downward,  and  therefore  the  field 
intensities  at  P  due  to  all  the  elements  of  the  wire  may  be  added 
algebraically.  The  field  intensity  at  P  due  to  the  current  in  the 
length  dl  is,  from  equation  (3), 

(I  sin  9}  dl 


and  therefore  the  total  field  intensity  at  P  due  to  all  the  elements 
of  the  wire  is 

(/  sin  0)  dl 


f/i 
H=\ 

J  -I 


x2 

2 

where  lt  =  OA  and  12  =  OB.  The  simplest  way  to  evaluate  this 
integral  is  to  express  the  different  variable  quantities  in  terms  of 
the  variable  angle  a  (see  Figure  52 ) .  From  the  figure  we  have 

T 

sin  0  =cos  a  •  x  =—      -  and  /  =r  tan  a.     Differentiating  the  latter 
cos   a 

expression  we  get  dl  = — ; — •      Let  at  be  the  numerical  value  of 
cos2  a 

the  angle  OP  A  and  a2  the  numerical  value  of  the  angle  OPE. 
Substituting  these  values  in  the  above  equation  we  get 

.  -.  (I  cos  a)  da     *  .  r1       *  I  \   (4\ 

H=  I  —  =-|  sin  a  I       =— I  sin  a^  +  sin  cu  I   *•  ' 


in  a2  ) 


All  quantities  in  this  equation  are  in  c.  g.  s.  units. 

When  the  wire  is  of  infinite  length  (or  practically,  when  the  dis- 
tance of  the  point  P  from  the  ends  of  the  wire  is  great  compared  to 
the  perpendicular  distance  r)  ar  and  a2  become  equal  to  90°,  and 
therefore  sin  a,=l  and  sin  a2  =  1 .  We  then  have  that  the  field 
intensity  at  a  distance  r  from  a  straight  wire  of  infinite  length 
carrying  a  current  I  is 


118  ELECTRICAL  ENGINEERING 

All  quantities  in  this  equation  are  in  c.  g.  s.  units.  The  direction 
of  the  field  intensity  is  perpendicular  to  the  plane  drawn  through 
the  wire  and  the  point;  the  lines  of  force  are  therefore  circles 
with  their  centers  along  the  wire  and  their  planes  perpendicular 
to  the  wire.  The  relation  between  the  direction  of  these  lines 
of  force  and  the  direction  of  the  current  is  conveniently  remem- 
bered by  the  rule  that  the  lines  of  force  are  in  the  direction  in 
which  a  right-handed  screw  must  be  turned  to  advance  it  in  the 
direction  of  the  current.  Equations  (4)  apply  to  a  variable  as 
well  as  to  a  continuous  current. 

The  above  formula  has  been  deduced  on  the  assumption  that 
the  wire  may  be  considered  as  a  geometrical  line.  It  can  also  be 
shown  that  the  field  intensity  at  any  point  P  at  a  distance  from  r 
from  a  long  circular  wire  of  finite  cross  section  is  also  given  by  the 
above  formula  provided  the  point  P  lies  outside  the  wire,  and  the 
current  density  is  uniform  over  the  section  of  the  wire ;  see  Article 
104.  At  any  point  inside  such  a  wire  the  field  intensity  is  also  per- 
pendicular to  the  plane  through  the  point  and  the  axis  of  the 
wire  but  is  equal  to 

2  /  r  (46) 

?'-rsr 

where  r  is  the  distance  of  the  point  from  the  center  of  the  wire,  a 
is  the  radius  of  the  wire  and  /  is  the  total  current  in  the  wire, 
and  all  quantities  are  in  c.  g.  s.  units.  Experiment  justifies 
assumption  that  the  current  density  in  a  wire  is  constant,  pro- 
vided the  wire  is  of  uniform  structure  and  the  current  is  a  con- 
tinuous current.  This  is  not  true  when  the  current  varies  rapidly 
with  time,  but  is  approximately  true  for  ordinary  variable  or 
alternating  currents  used  in  practical  work,  provided  the  wire 
is  non-magnetic  and  has  a  diameter  less  than  one  inch.  The 
above  formula  for  the  field  intensity  outside  a  wire  is  also  ap- 
proximately true  for  a  wire  which  has  a  cross  section  of  any  shape, 
provided  the  distance  of  the  point  from  the  wire  is  great  com- 
pared to  the  greatest  diameter  of  the  wire. 

74.  Magnetic  Field  Due  to  an  Electric  Current  in  a  Circular 
Coil  of  Wire.  —  Another  useful  application  of  equation  (3)  is  the 
calculation  of  the  intensity  of  the  magnetic  field  at  any  point  due 
to  a  current  in  a  circular  coil  of  wire.  The  solution  of  this  problem 
except  for  points  on  the  axis  of  the  coil  is  quite  difficult;  the 
solution  for  a  point  on  the  axis  of  the  coil  is  obtained  as  follows. 


CONTINUOUS  ELECTRIC  CURRENTS 


119 


Let  the  coil  have  but  a  single  turn  and  let  us  consider  the  wire  form- 
ing the  coil  as  a  geometrical  line  making  a  circle  of  radius  r.  Let  P 
be  any  point  on  the  axis  of  this  coil  at  a  distance  a  from  the  plane 
of  the  coil.  Let  I  be  the  current  in  the  wire,  dl  any  elementary 
length  in  the  circumference  of  the  wire  and  dl'  an  equal  elementary 
length  diametrically  opposite  dl.  The  current  in  dlf  will  be  in  the 
opposite  direction  from  that  in  dl.  In  Fig.  53  ISt  the  current  be 

dl 


Fig.  53. 

up  toward  the  reader  at  dl  and  down  at  dl'.  (The  standard  con- 
vention for  showing  a  current  coming  up  is  a  circle  with  a  dot 
in  it,  and  for  a  current  going  down  a  circle  with  a  cross  in  it.) 

The  field  intensity  at  P  due  to  the  current  in  dl  is  then,  from 
equation  (3), 

Idl 


since  the  line  drawn  from  P  to  dl  is  perpendicular  to  dl,  whence  in 
equation  (3)  sin  0  =  1.  This  field  intensity  is  perpendicular  to  the 
plane  through  dl  and  P,  and  is  therefore  in  the  plane  of  the  paper 
in  Fig.  53,  perpendicular  to  the  line  from  dl  to  P  in  the  direction 
indicated.  This  field  intensity  dH  may  be  resolved  into  two 
components,  one  parallel  to  the  axis  OP  of  the  coil,  and  the  other 
perpendicular  to  OP.  Similarly,  the  current  in  dl'  produces  an 
equal  field  intensity  dH'  perpendicular  to  the  line  from  dl'  to  P, 
which  may  also  be  resolved  into  two  components,  one  parallel  to 
and  the  other  perpendicular  to  OP.  The  perpendicular  com- 
ponents due  to  dl  and  dl'  are  equal;  similarly  for  any  other  pair 
of  equal  elementary  lengths  diametrically  opposite  in  the  circum- 
ference of  the  circle.  Hence  the  resultant  field  intensity  at  P  is 
the  sum  of  the  components  parallel  to  OP  of  the  field  inten- 


120  ELECTRICAL  ENGINEERING 

sities  at  P  due  to  all  the  elementary  lengths  in  the  circumference 
of  the  circle.  The  component  parallel  to  OP  of  the  field  intensity 
dH  due  to  any  element  is 

ju      •  Idl  r 

dH.  sina= 


which,  integrated  along  the  circumference  of  the  circle,  gives  as 
the  value  of  the  resultant  field  intensity  at  P* 


"J: 


-f  '  (5) 

i    -*»«  \    A,  [  Cl"  "J^  fy**/  ]    £ 

1=0 

All  quantities  in  this  equation  are  in  c.  g.  s.  units. 

The  direction  of  the  field  intensity  at  any  point  along  the  axis 
is  the  direction  in  which  a  right-handed  screw  placed  at  this 
point  advances  when  it  is  turned  in  the  direction  of  the  current. 

The  field  intensity  at  the  center  of  the  coil  is  found  by  putting 
a  equal  to  zero  nr  the  above  equation,  which  gives 

H  Jit*  (5a) 

r 

The  field  intensity  at  any  point  on  the  axis  of  a  circular  coil 
which  has  any  number  of  concentric  turns  may  also  be  calculated 
from  equation  (5),  by  calculating  the  intensity  due  to  each  turn 
separately  and  adding  these  separate  intensities.  In  the  case  of 
a  circular  coil  with  a  concentrated  winding  of  N  turns,  i.e.,  when 
the  N  turns  are  so  close  together  that  they  may  all  be  considered 
as  occupying  but  a  single  geometrical  line,  the  field  intensity  at 
the  center  of  the  coil  is,  from  (5a), 

„      27TNI  (56) 

rlc=  — 

r 

The  winding  of  a  coil  may  be  considered  as  concentrated  when 
the  radius  of  the  coil  is  large  compared  with  the  linear  dimensions 
of  the  cross  section  of  the  winding. 

In  Article  108  is  deduced  by  a  different  method  the  field  in- 
tensity for  a  point  inside  a  long  coil  wound  in  the  form  of  a  long 
cylindrical  helix  or  "  solenoid." 

75.  Absolute  Measurement  of  an  Electric  Current.  —  When  a 
current  is  established  in  an  insulated  wire  wound  into  a  circular 
coil  of  N  turns,  the  strength  of  this  current  can  be  determined 
experimentally  in  terms  of  quantities  which  can  be  measured  or 


CONTINUOUS   ELECTRIC  CURRENTS 


121 


calculated.  Let  such  a  coil  (Fig.  54)  be  set  up  with  the  plane  of  its 
windings  parallel  to  the  direction  of  the  earth's  magnetic  field 
and  let  a  small  magnetic  needle  be  suspended  at  the  center  of  the 
coil  in  such  a  manner  that  it  is  free  to  turn  about  a  vertical  axis. 
When  there  is  no  current  in  the  coil,  this  needle  will  then  point  in 
a  direction  parallel  to  the  plane  of  the  .coil.  When  a  current  is 
established  in  the  coil,  this  current  will  set  up  a  magnetic  field  at 
right  angles  to  the  plane  of  the  coil,  and  therefore  the  direction 
of  the  resultant  field  at  the  center  of  the  coil  will  be  changed  and 
the  needle  will  .therefore  be  deflected  a  certain  angle  ff.  This 
angle  will  be  equal  to  the  angle  between  the  direction  of  the  result- 
ant field  and  the  direction  of  the  horizontal  component  of  the 
earth's  field  at  the  center  of  the  coil.  This  latter  field  intensity, 
which  we  may  call  He,  can  be  determined  by  the  method  described 
in  Article  42.  The  intensity  of  the  field  at  the  center  of  the  coil 
Hc  is  given  by  equation  (56). 


H, 


We  then  have  that 


Fig.  54 


,      a     Hc    2-rrNI 

tan  6= — -= . 


or 


rHetan0 


(6) 


27T  N 

where  all  the  quantities  are  in  c.  g.  s.  units. 


Since  all  the  quantities  in  the  right-hand  member  of  this  equa- 


122  ELECTRICAL  ENGINEERING 

tion  can  be  measured,  the  strength  of  the  current  /  can  be  cal- 
culated. Such  a  device  is  called  a  tangent  galvanometer.  The 
accuracy  of  the  instrument  depends  upon  the  accuracy  with  which 
the  horizontal  component  of  the  earth's  field  may  be  measured,  but 
an  accurate  measurement  of  the  latter  is  difficult.  Besides,  in  any 
ordinary  laboratory  or  testing  room  the  electric  circuits  in  the 
building  and  the  surrounding  streets  also  produce  magnetic 
fields  which  act  on  the  needle,  and  these  fields  are  continually 
changing.  Such  an  instrument  is  therefore  seldom  used  now- 
adays. 

A  much  more  accurate  method  of  determining  the  absolute  value 
of  an  electric  current  is  to  cause  the  same  current  to  flow  through 
two  parallel  coils,  one  of  which  is  suspended  from  one  arm  of  a 
delicate  balance.  The  stationary  coil  then  produces  a  magnetic 
field  which  produces  a  pull  on  the  movable  coil  and  the  amount  of 
this  pull  can  be  readily  measured.  From  equations  (2)  and  (3) 
the  amount  of  this  pull  in  terms  of  the  current  and  the  dimensions 
of  the  coils  can  also  be  deduced,  and  since  both  the  pull  and  the 
dimensions  of  the  coil  can  be  measured,  the  strength  of  the 
current  can  be  calculated.  Such  an  instrument  is  called  an  " ab- 
solute" current  balance.  This  is  also  the  principle  of  the  Kelvin 
current  balances;  in  the  latter,  however,  the  current  strength  is  not 
determined  directly  from  the  pull  and  the  dimensions  of  the  coils, 
but  is  calculated  from  the  position  of  a  "  rider  "  on  an  arbitrary 
scale;  the  latter  is  "  calibrated  "  by  comparison  either  with  an 
absolute  balance  or  with  a  silver  voltameter  (see  Article  79). 

76.  Comparison  of  the  Strengths  of  Electric  Currents. —  Gal- 
vanometers and  Ammeters.  —  We  have  just  seen  how  the  strength 
of  an  electric  current  flowing  through  the  coils  of  a  current  balance 
may  be  measured  in  terms  of  the  dimensions  of  the  coils  and  the 
pull  of  the  fixed  coil  on  the  movable  coil.  The  accurate  measure- 
ment of  an  electric  current  by  this  method  requires  a  balance  con- 
structed with  great  care  and  numerous  precautions  have  to  be 
taken  in  using  it.  Much  simpler  and  cheaper  instruments  can 
be  constructed  which  will  indicate  by  the  deflection  of  a  needle 
or  spot  of  light  the  relative  magnitude  of  the  electric  currents  which 
may  be  established  in  them. 

The  simplest  form  of  such  an  instrument  is  a  device  which 
consists  essentially  of  a  magnet  suspended  inside  a  coil  of  insulated 
wire  which  may  be  connected  in  series  with  the  conductor  in  which 


CONTINUOUS  ELECTRIC  CURRENTS  123 

the  current  to  be  measured  is  established.  The  magnetic  field 
produced  inside  the  coil  by  the  current  produces  a  force  on 
the  needle  and  causes  it  to  deflect  from  its  position  of  equilibrium; 
this  force,  and  therefore  the  deflection,  varies  with  the  current  in 
the  coil.  A  device  of  this  kind  is  called  a  moving  needle,  or  Thom- 
son, galvanometer.  The  value  of  the  current  corresponding  to 
a  given  deflection  can  be  determined  once  for  all  (provided  the 
conditions  of  operation  remain  unchanged)  by  connecting  in  series 
with  the  galvanometer  a  standard  current  balance  and  noting 
the  value  of  the  deflection  corresponding  to  various  current 
strengths  as  indicated  by  the  standard  balance;  that  is,  the  gal- 
vanometer can  be  "  calibrated  "  by  comparing  it  directly  with 
the  standard  balance.  In  practice,  instead  of  using  a  standard 
balance,  one  ordinarily  uses  a  "  secondary  "  standard,  that  is, 
some  other  instrument  which  has  been  previously  calibrated  by 
comparison  with  a  "  primary  "  or  absolute  standard. 

The  needle  galvanometer  just  described  does  not  "  hold  "  its 
calibration  for  any  great  length  of  time,  on  account  of  the  effects 
of  temperature,  moisture,  etc.,  on  the  fiber  supporting  the  mag- 
netic needle  and  also  on  account  of  the  variation  in  the  direction 
and  intensity  of  the  earth's  magnetic  field.  A  superior  form  of 
galvanometer  for  most  practical  purposes  consists  of  a  coil  of  fine 
insulated  wire  suspended  by  a  metal  fiber  or  thin  metal  strip,  be- 
tween the  two  poles  of  a  powerful  permanent  magnet  made  in 
the  form  of.  a  horseshoe.  The  fixed  end  of  the  metal  strip  sup- 
porting the  coil  is  connected  to  a  suitable  terminal  or  "  binding 
post  "  on  the  frame  of  the  instrument  and  the  other  end  to  one  end 
of  the  wire  forming  the  coil.  The  other  end  of  the  coil  is  con- 
nected through  a  second  metal  strip,  usually  wound  in  the  form 
of  a  light  spiral  spring,  to  a  second  terminal  or  binding  post.  When 
the  two  terminals  of  such  an  instrument  are  connected  in  series 
with  the  circuit  in  which  the  current  to  be  measured  is  established, 
the  same  current  is  also  established  in  the  wire  forming  the 
movable  coil,  and  the  force  produced  on  this  coil  by  the  magnetic 
field  due  to  the  permanent  magnet  causes  a  deflection  of  the  coil. 
(See  equation  2.)  This  deflection  can  be  read  by  noting  the 
deflection  of  a  spot  of  light  reflected  from  a  mirror  attached  to 
the  coil.  This  type  of  galvanometer  is  called  a  moving  coil,  or 
D'Arsonval,  galvanometer,  from  the  name  of  the  physicist  who 
introduced  it.  A  D'Arsonval  galvanometer  can  be  calibrated 


124  ELECTRICAL  ENGINEERING 

by  comparing  it  with  a  standard  instrument  in  the  same 
way  that  a  moving  needle,  or  Thomson,  galvanometer  is 
calibrated. 

An  instrument  extensively  used  for  the  measurement  of 
electric  currents,  and  known  as  the  Weston  ammeter,  is  essen- 
tially a  D' Arson  val  galvanometer.  In  this  instrument  the 
movable  coil,  instead  of  being  suspended  by  a  metal  strip,  is  pro- 
vided with  a  fine  pivot  which  rests  in  a  jewel  bearing.  The 
current  is  "  led  to  and  from  "  the  instrument  by  means  of  two 
small  flat  spiral  springs  connected  to  the  two  ends  of  the  coil 
respectively  and  also  to  the  terminals  on  the  outside  of  the  case 
of  the  instrument.  These  springs  also  serve  to  hold  the  coil 
normally  in  a  definite  position.  A  light  metal  pointer  is  attached 
to  the  coil,  and  the  position  of  the  pointer  is  read  off  on  a  scale 
which  is  marked  to  read  directly  in  amperes.  The  scale  of  such 
an  instrument  is  seldom  exactly  correct,  and  for  accurate  work 
the  instrument  must  be  calibrated  in  the  manner  described  above. 
There  are  other  makes  of  ammeters  based  upon  the  principle 
of  the  D'Arsonval  galvanometer,  and  still  others  in  which  the 
current  in  the  coil  of  the  instrument  sets  up  a  magnetic  field  which 
exerts  a  force  on  a  piece  of  soft  iron  and  thereby  causes  the  de- 
flection of  a  pointer  over  a  graduated  scale.  For  a  detailed 
description  of  ammeters,  the  reader  is  referred  to  any  text-book 
on  electric  measuring  instruments. 

77.  Electrolysis  and  Electrolytes.  —  Having  now  seen  how 
the  strength  of  an  electric  current  may  be  measured,  we  are  ready 
to  investigate  some  of  the  other  phenomena  which  are  attributed 
to  the  flow  of  an  electric  current.  An  important  group  of  phenom- 
ena are  those  which  take  place  in  a  conducting  solution  when  a 
current  is  established  through  the  solution.  In  the  first  place,  in 
any  solution  which  is  a  conductor,  chemical  decomposition  always 
takes  place.  The  change  that  takes  place  in  the  copper  sulphate 
solution  described  in  Article  62  is  an  illustration.  Any  substance 
the  constituents  of  which  are  separated  when  an  electric  current 
is  established  in  it,  is  called  an  electrolyte,  and  the  process  of  sepa- 
ration is  called  electrolysis.  All  conducting  liquids  other  than 
molten  metals  are  electrolytes;  gases,  when  they  become  conduct- 
ing, are  also  electrolytes.  Some  solids  are  also  electrolytes,  an  exam- 
ple of  which  is  silver  iodide.  The  decomposition  that  takes  place  in 
the  electrolyte  is  found  to  be  confined  entirely  to  the  portions  of 


CONTINUOUS   ELECTRIC  CURRENTS  125 

the  electrolyte  in  contact  with  the  metal  plates  or  wires  (called  the 
electrodes)  which  connect  the  electrolyte  to  the  rest  of  the  circuit. 
The  results  of  the  decomposition  are  therefore  usually  said  to  be 
deposited  at  the  electrodes;  though,  of  course,  if  the  substance 
liberated  is  a  gas,  it  will  immediately  escape  through  the  surface 
of  the  liquid,  or  if  the  substance  liberated  is  soluble,  it  will  go  into 
solution,  or  if  it  is  a  solid  which  is  not  soluble,  it  may  fall  to  the 
bottom  of  the  vessel  containing  the  electrolyte.  In  certain  cases 
when  the  result  of  the  decomposition  is  a  metal, the  metal  becomes 
firmly  attached  to  the  electrode  where  it  is  liberated.  The  copper 
deposited  on  the  copper  wire  in  the  copper  sulphate  solution  in 
Article  62  is  an  example.  The  important  industry  of  electroplat- 
ing is  based  upon  this  fact. 

The  electrode  through  which  the  current  enters  the  solution 
is  called  the  anode,  and  the  electrode  through  which  the  current 
leaves  the  solution  is  called  the  cathode.  With  very  few  exceptions, 
an  element,  or  such  a  group  of  elements  as  is  called  a  "  radical," 
is  always  deposited  at  the  same  electrode,  no  matter  from  what 
compound  it  may  be  liberated.  Hydrogen  and  metals  are  always 
deposited  at  the  cathode. 

As  the  result  of  a  series  of  careful  experiments  Faraday  found 
that  a  simple  relation  exists  between  the  rate  at  which  a  sub- 
stance is  deposited  from  an  electrolyte, 'when  an  electric  current 
is  established  in  it,  and  the  strength  of  the  current.  This  rela- 
tion is 

1.  The  rate  (mass  per  unit  time)  at  which  a  substance  is  de- 
posited from  an  electrolyte,  when  an  electric  current  is  established 
in  it,  is  directly  proportional  to  the  strength  of  the  current  estab- 
lished. 

Faraday  also  found  that  a  simple  relation  exists  between  the 
masses  of  the  various  substances  deposited,  when  the  same  current 
is  established  in  several  electrolytes,  and  the  chemical  equiva- 
lents* of  these  substances.  This  relation  is 

2.  When  the  same  current  is  established  through  different 
*The  chemical  equivalent  of  an  element  or  radical  is  the  atomic  weight 

of  the  element  or  radical  divided  by  its  valence.  By  valence  is  meant  the 
number  of  atoms  of  hydrogen  with  which  one  atom  of  the  element  or  radi- 
cal will  form  a  stable  combination.  For  example,  the  valence  of  oxygen  is 
2,  since  2  atoms  of  hydrogen  combine  with  1  atom  of  oxygen  to  form 
water.  The  valence  of  copper  in  cuprous  compounds  is  1  and  in  cupric 
compounds  2. 


126  ELECTRICAL  ENGINEERING 

electrolytes  the  rates  (mass  per  unit  time)  at  which  the  various 
substances  are  deposited  are  directly  proportional  to  the  chemical 
equivalents  of  these  substances. 

These  two  statements  of  experimental  facts  are  known  as 
Faraday 's  Laws  of  Electrolysis. 

78.  Electrochemical  Equivalent  of  a  Substance.  —  Faraday's 
first  law  may  be  expressed  mathematically  by  the  equation 

"=*/  (7) 

t 

where  7  is  the  current  established  in  the  electrolyte,  m  the 
mass  of  a  given  substance  deposited  in  time  t,  and  k  a  constant 
of  proportionality,  the  value  of  which  depends  only  upon  the 
nature  of  the  substance  and  the  units  in  which  m,  t  and  /  are 
measured.  This  constant  is  called  the  electrochemical  equivalent 
of  the  substance.  Its  value  for  any  substance  may  be  readily 
determined  experimentally  by  measuring  the  current  7  by  means 
of  an  absolute  current  balance,  and  measuring  the  mass  of  the 
substance  deposited  when  this  current  flows  through  the  electrolyte 
for  a  given  length  of  time.  The  values  of  the  electrochemical 
equivalent  for  silver,  copper,  hydrogen  and  oxygen,  when  m  is 
measured  in  grams,  t  in  seconds  and  7  in  amperes,  are 
Silver  >  0.001118 

Copper  (from  cupric  solutions)  0.0003293 

Hydrogen  0.00001044 

Oxygen  0.0000829 

In  equation  (7)  the  current  is  assumed  to  be  continuous ;  when 
the  current  varies  with  time  the  mathematical  expression  of 
Faraday's  first  law  is 

dm 

—  =fci 
dt 

or  the  total  mass  of  the  substance  deposited  in  time  t  is 

ft 

m=k  I   idt  (la) 

J  o 

79.  The  International  Ampere.  —  Knowing  the  value  of  the 
electrochemical  equivalent  of  a  given  substance,  for  example, 
silver,  one  can  readily  determine  the  strength  of  the  current  flow- 
ing through  an  electrolyte  from  which  this  substance  is  deposited, 
by  measuring  the  number  of  grams  of  this  substance  deposited 


CONTINUOUS  ELECTRIC  CURRENTS  127 

in  a  measured  interval  of  t  seconds.  The  only  apparatus  required 
is  a  suitable  receptacle  for  holding  the  electrolyte  and  suitable 
electrodes  for  leading  the  current  in  and  out.  Such  a  piece  of 
apparatus  is  called  a  voltameter.  From  the  ease  and  accuracy 
with  which  these  measurements  can  be  made,  the  International 
Congress  of  Electricians  (Chicago,  August  21,  1893)  adopted  the 
following  as  the  definition  of  the  ampere: 

"As  a  unit  of  current  (shall  be  taken),  the  International 
Ampere,  which  is  one-tenth  of  the  unit  of  current  of  the  C.  G.  S. 
system  of  electromagnetic  units,  and  which  is  represented  suffi- 
ciently well  for  practical  use  by  the  unvarying  current,  which, 
when  passed  through  a  solution  of  nitrate  of  silver  in  water, 
in  accordance  with  the  accompanying  specification  (A),  deposits 
silver  at  the  rate  of  0.001118  gramme  per  second." 

The  specification  A  referred  to  describes  in  detail  the  con- 
struction of  the  voltameter  and  the  method  of  using  it.  This 
specification  will  be  found  in  full  in  Foster's  Electrical  Engineer's 
Pocket  Book,  page  10.  The  above  definition  of  the  ampere  has 
been  legalized  by  most  of  the  civilized  countries,*  and  is  therefore 
sometimes  called  the  "  legal "  definition  of  the  ampere. 

80.  Quantity  of  Electricity.  —  In  the  case  of  a  current  of  water 
flowing  through  a  pipe,  the  strength  of  the  water  current  is  defined 
as  the  quantity  of  water  which  flows  across  any  cross  section  of 
the  pipe  in  unit  time.  In  the  case  of  what  is  called  a  current 
of  electricity,  we  have  found  it  convenient  to  define  first 
the  strength  of  the  electric  current.  From  the  analogy  which  is 
found  to  exist  between  the  properties  which  must  be  attributed 
to  the  something  called  electricity  and  the  properties  of  an 
incompressible  fluid,  the  quantity  Q  of  electricity  flowing  across 
any  cross  section  of  a  conductor  in  any  time  t  may  be  defined 
as  the  product  of  the  strength  of  the  current  /  at  this  cross  section 
by  the  time  /,  that  is, 

Q=It  (8) 

provided  the  current  is  a  continuous  current,  i.e.,  does  not  vary 
with  time.  In  case  the  current  does  vary  with  time  the  general 
definition  of  the  quantity  Q  of  electricity  which  flows  across 
any  given  cross  section  of  the  conductor  in  any  interval  of  time  t, 

*Germany  and  Switzerland  are  exceptions;  in  these  countries  the  electro- 
chemical definition  is  taken  as  the  fundamental  definition  of  current  strength. 


128  ELECTRICAL  ENGINEERING 

is  the  integral  with  respect  to  time  of  the  variable  current  at 
that  section,  that  is 


Q=[idt  (8a) 

J  o 


where  i  represents  the  value  of  the  current  at  any  instant,  dt  an 
infinitesimal  interval  of  time  measured  from  this  instant,  and  t 
the  total  time  during  which  the  current  flows.  From  the  relation 
expressed  by  equation  (7 a)  the  quantity  of  electricity  which 
flows  through  an  electrolyte  in  any  interval  of  time  may  be 
determined  by  measuring  the  mass  of  the  substance  deposited 
at  either  electrode  in  this  interval.  In  Article  109  is  described 
the  method  usually  employed  for  measuring  the  quantity  of 
electricity  corresponding  to  a  variable  current. 

Since  in  the  case  of  a  continuous  current  the  total  current 
strength  is  the  same  at  all  cross  sections  of  a  given  conductor,  it 
follows  that  the  quantity  of  electricity  flowing  across  each  cross 
section  of  the  conductor  in  any  given  interval  of  time  must  also 
be  the  same  at  each  cross  section.  Note  the  analogy  with  an 
incompressible  fluid.  Hence  the  flow  of  a  current  of  strength  7 
through  the  conductor  for  t  seconds  may  be  looked  upon  as 
equivalent  to  the  transfer  of  / 1  units  of  electricity  from  one  end 
of  the  conductor  to  the  other,  just  as  a  current  of  ten  cubic  feet 
of  water  per  second  in  a  pipe  for  five  seconds  is  equivalent  to  a 
transfer  of  10  x  5  =50  cubic  feet  of  water  from  one  end  of  the  pipe 
to  the  other.  In  the  case  of  water  in  a  pipe  the  50  cubic-  feet  of 
water  which  enter  the  pipe  may  not  be  the  same  as  the  50  cubic 
feet  which  leave  the  pipe  at  the  other  end.  In  the  same  way,  it 
is  not  necessary  that  we  look  upon  the  electricity  which  enters 
one  end  of  a  conductor  as  being  the  same  electricity  as  leaves  the 
conductor  at  its  other  end. 

The  unit  of  quantity  of  electricity  in  the  c.  g.  s.  electromag- 
netic system  of  units  is  the  quantity  of  electricity  transferred  by 
a  current  of  one  abampere  for  one  second.  This  unit  is  called 
the  absolute  unit  of  quantity  or  the  abcoulomb.  The  practical 
unit  of  quantity  of  electricity  is  the  coulomb,  which  was  defined 
by  the  International  Congress  of  Electricians  as  follows: 

"  As  the  unit  of  quantity  (shall  be  taken),  the  International 
Coulomb,  which  is  the  quantity  of  electricity  transferred  by  a 
current  of  one  international  ampere  in  one  second." 

The  ampere-hour,  i.e.,  the  quantity  of  electricity  corresponding 


CONTINUOUS  ELECTRIC  CURRENTS  12£> 

to  a  current  of  one  ampere  for  one  hour  is  also  employed  in  practice. 

In  accordance  with  these  definitions  we  then  have 
1  abcoulomb   =10  coulombs 
1  ampere-hour  =3600  coulombs 

81.  Electric  Resistance. — Joule's  Law. — One  of  the  phe- 
nomena always  associated  with  an  electric  current  in  a  conductor 
is  the  dissipation  of  heat  energy  in  the  conductor.  In  the  case 
of  a  wire  of  uniform  structure  kept  at  a  constant  uniform  tempera- 
ture experiment  shows  that  the  rate  at  which  heat  energy  is  dis- 
sipated in  a  given  length  of  the  wire,  between  any  two  points  1 
and  2  say,  when  a  continuous  current  is  established  in  the  wire, 
is  directly  proportional  to  the  square  of  the  strength  of  the  current 
in  this  wire.  That  is,  calling  Ph  the  rate  at  which  heat  energy 
is  developed  in  the  wire  between  the  points  1  and  2,  and  7  the 
strength  of  the  current  in  this  portion  of  the  wire,  then 

Pk  =  RP  (9) 

where  R  is  a  constant  depending  upon  the  dimensions  and  tem- 
perature of  the  wire,  the  nature  of  the  substance  forming  the  wire, 
and  the  units  in  which  Ph  and  7  are  measured,  but  is  independent 
of  the  current  strength.  This  factor  R  is  called  the  resistance  of 
the  wire,  and  the  statement  of  the  experimental  fact  represented 
by  this  equation  is  called  Joule's  Law,  from  the  name  of  the 
scientist  who  first  clearly  enunciated  the  fact.  The  resistance 
of  a  given  length  of  wire  may  then  be  defined  as  the  ratio  of  the 
rate  at  which  a  continuous  current  produces  heat  energy  in  the  wire 
to  the  square  of  the  strength  of  this  current.  When  the  rate  Ph  at 
which  the  heat  energy  is  dissipated  is  expressed  in  ergs  per  second 
and  the  current  7  in  abamperes,  a  resistance  of  one  unit  is  equal  to 
the  ratio  of  one  erg  per  second  to  one  abampere-squared.  This 
unit  of  resistance  is  called  the  c.  g.  s.  electromagnetic  unit  of  resist- 
ance or  the  abohm.  When  the  rate  Ph  at  which  heat  energy  is  dis- 
sipated is  expressed  in  joules  per  second  (one  joule  by  definition  is 
equal  to  107  ergs)  and  the  current  7  in  amperes,  a  resistance  of  one 
unit  is  equal  to  the  ratio  of  one  joule  per  second  to  one  ampere- 
squared.  This  unit  is  called  the  practical  unit  of  resistance  or  the 
ohm.  The  relation  between  the  ohm  and  the  abohm  is  therefore 
1  ohm=109  abohms. 

To  express  small  resistances  a  unit  one-millionth  of  the  size 
of  an  ohm  is  ordinarily  used;  this  unit  is  called  the  microhm.  To 
express  large  resistances  a  unit  one  million  times  the  size  of  an 


130  ELECTRICAL  ENGINEERING 

ohm  is  frequently  used;  this  unit  is  called  the  megohm.     Hence 
1  ohm         =108  microhms 
1  megohm  =  10°  ohms. 

The  resistance  of  a  wire  to  a  variable  current  is  the  same  as 
its  resistance  to  a  continuous  current  provided  the  wire  is  small 
and  the  current  does  not  vary  rapidly  with  time.  In  the  case  of 
a  large  wire  the  resistance  of  any  current  filament  (see  Article  101) 
is  the  same  to  the  variable  as  to  a  continuous  current,  but  the  re- 
sistance of  the  wire  as  a  whole  is  greater.  The  rapid  variation 
of  the  current  with  time  causes  a  different  distribution  of  the 
stream  lines  of  the  current  and  thereby  produces  a  greater  heating 
than  that  which  takes  place  due  to  a  continuous  current.  See 
Article  121. 

82.  Absolute  Measurement  of  Electric  Resistance.  —  The 
determination  of  the  electric  resistance  in  terms  of  the  quantities 
specified  in  the  above  definition  would  require  the  measurement 
of  the  heat  energy  dissipated  in  the  wire  in  a  given  interval  of 
time  when  a  continuous  electric  current  of  known  strength  is 
established  in  the  wire  and  the  wire  kept  at  constant  temperature. 
The  interval  of  time  can  be  readily  measured,  and  the  current 
strength  may  be  determined  directly  by  an  absolute  current  bal- 
ance or  by  means  of  any  kind  of  galvanometer  or  ammeter  which 
has  been  calibrated.  The  heat  energy  dissipated  in  the  wire 
could  be  determined  by  some  calorimetric  measurement.  Calori- 
metric  measurements,  however,  are  difficult  and  at  best  are  not 
susceptible  of  a  high  degree  of  accuracy.  A  much  more  accurate 
method  of  measuring  the  electric  resistance  of  a  wire  in  terms  of 
quantities  which  may  be  measured  or  readily  calculated,  is  the 
following,  which  is  based  upon  the  principle  of  electromag- 
netic induction  (see  Chapter  IV),  but  the  only  quantities  to  be 
measured  are  those  which  have  already  been  defined.  A  metal 
disc  D,  mounted  on  a  metal  axis  coincident  with  the  axis  of  a  coil 
of  wire  C,  is  arranged  so  that  it  can  be  rotated  with  a  constant 
angular  velocity  CD.  The  coil  C  and  the  resistance  R  to  be  meas- 
ured are  connected  in  series  with  a  battery  B.  A  galvanometer 
G  has  one  of  its  terminals  connected  by  a  wire  to  the  axis  of  the 
disc  and  the  other  terminal  to  one  end  of  the  resistance  R,  the  other 
end  of  R  is  connected  by  a  wire  to  a  metal  "  brush  "  making 
contact  with  the  rim  of  trie  disc.  We  have  then  two  closed  circuits 
in  each  of  which  the  resistance  R  forms  a  part;  one  circuit  is  the 


CONTINUOUS  ELECTRIC  CURRENTS 


131 


battery,  the  coil  and  the  resistance  R  and  the  connecting  wires, 
and  the  other  circuit  is  the  galvanometer,  the  disc,  the  resistance 
R  and  the  connecting  wires.  When  the  disc  is  set  in  rotation,  it 
is  found  that  the  current  through  the  galvanometer  depends  upon 
the  angular  speed  at  which  the  disc  is  driven  (due  to  the  cutting 
by  the  disc  of  the  lines  of  induction  produced  by  the  current  in  the 
coil,  see  Chapter  IV),  and  as  a  result  of  the 'two  principles  known 
as  KirchhofFs  Laws  (see  Article  98),  it  can  be  deduced  that  when 
the  current  in  the  galvanometer  is  zero,  the  relation  between 


C 


Fig.  55. 


the  value  of  the  resistance  R  as  above  defined,  in  abohms,  and 
the  angular  speed  in  radians  per  second,  must  be 


-rr 


(10) 


where  M  represents  the  number  of  lines  of  induction  threading 
the  disc  D  due  to  the  magnetic  field  set  up  by  a  current  of  one  ab- 
ampere  in  the  coil  C.  This  quantity  M  can  be  calculated  in  terms 
of  the  dimensions  of  the  coil,  which  can  of  course  be  measured, 
as  can  also  the  angular  speed  a).  This  method  for  determining 
the  value  of  a  resistance  is  known  as  Lorenz's  Method.  There 
are  still  other  methods  for  determining  experimentally  the  value 
of  a  resistance  in  terms  of  quantities  which  can  be  actually  meas- 
ured or  calculated,  some  of  which  are  described  in  J.  J.  Thomson's 
Elements  of  Electricity  and  Magnetism,  page  462  ff. 


132  ELECTRICAL  ENGINEERING 

The  results  of  a  large  number  of  measurements  by  these  various 
methods  show  that  a  column  of  mercury  106.3  centimeters  long 
which  has  a  uniform  cross  section  of  1  square  millimeter  has  a  resist- 
ance of  one  ohm  at  zero  degrees  centigrade.  A  column  of  mercury 
of  these  dimensions  has  a  mass  of  14.4521  grams.  Hence  the 
adoption  of  the  following  definition  of  the  ohm  by  the  International 
Congress  of  Electricians : 

"As  a  unit  of  resistance  (shall  be  taken),  the  International 
Ohm,  which  is  based  upon  the  ohm  equal  to  109  units  of  resistance 
of  the  C.  G.  S.  system  of  electromagnetic  units,  and  is  represented 
by  the  resistance  offered  to  an  unvarying  electric  current  by  a 
column  of  mercury  at  a  temperature  of  melting  ice,  14.4521 
grammes  in  mass,  of  a  constant  cross  sectional  area,  and  of  the 
length  106.3  centimeters." 

This  definition,  like  that  of  the  ampere,  has  been  legalized  by 
most  civilized  countries. 

Although  the  absolute  measurement  of  a  resistance  is  com- 
paratively difficult,  the  comparison  of  the  values  of  two  or  more 
resistances  is  a  very  simple  matter  and  can  be  carried  out  with  a 
high  degree  of  accuracy.  All  these  methods  of  comparison  are 
based  upon  KirchhofFs  Laws.  (See  Article  92.) 

83.  Specific  Resistance  or  Resistivity.  —  As  already  noted,  the 
resistance  of  a  conductor  depends  upon  its  dimensions.  It  is 
found  by  experiment  that  the  resistance  of  a  conductor  of  uniform 
cross  section  throughout  its  length,  when  the  conductor  is  kept 
at  a  uniform  temperature  throughout  and  the  current  density 
(see  Article  101)  is  uniform  over  its  cross  section,  varies  directly 
as  the  length  of  the  conductor  and  inversely  as  the  area  of  its 
cross  section  (perpendicular  to  its  length),  but  does  not  depend 
upon  the  shape  of  its  cross  section.  That  is,  calling  I  the  length 
of  the  conductor,  A  the  area  of  its  cross  section,  and  R  the  resist- 
ance of  this  conductor,  then 

r,  I  (11) 

R=PA 

where  p  is  a  factor  of  proportionality  which  depends  upon  the 
material  of  which  the  conductor  is  made  and  the  temperature  at 
which  the  conductor  is  kept,  and  also  upon  the  units  in  which 
R,  I,  and  A  'are  expressed.  The  value  of  this  factor  p  for  any 
conductor,  at  any  temperature,  is  called  the  specific  resistance 
or  the  resistivity  of  the  conductor  at  that  temperature.  When  R 


CONTINUOUS  ELECTRIC  CURRENTS  133 

is  expressed  in  microhms,  I  in  centimeters  and  A  in  square  centi- 
meters, this  factor  p  is  equal  to  the  resistance  in  microhms  of 
a  cube  of  the  conductor  1  cm.  on  each  edge,  provided  the  stream 
lines  of  the  current  are  parallel  to  four  parallel  edges  of  the  cube 
and  the  current  density  over  the  section  at  right  angles  to  these 
streamlines  is  uniform  (see  Article  101).  The  specific  resistance 
of  the  conductor  may  then  be  expressed  as  so  many  microhms 
per  centimeter-cube.  Similarly,  when  R  is  expressed  in  microhms, 
/  in  inches  and  A  in  square  inches,  this  factor  p  is  equal  to  the 
resistance  in  microhms  of  an  inch-cube  of  the  conductor;  the 
specific  resistance  of  the  conductor  may  then  be  expressed  as  so 
many  microhms  per  inch-cube.  Again,  when  R  is  expressed  in 
ohms,  I  in  feet  and  A  in  circular  mils*  this  factor  p  is  equal  to 
the  resistance  in  ohms  of  a  portion  of  the  conductor  one  foot  long 
and  one  circular  mil  in  cross  section;  the  specific  resistance  of  a 
conductor  may  then  be  expressed  as  so  many  ohms  per  mil-foot.  ' 
There  is  still  another  way  of  expressing  the  specific  resistance 
of  a  conductor,  which  was  formerly  much  used  and  is  still  em- 
ployed sometimes  in  wire  specifications.  The  cross  section  of  a 
bar  or  wire  of  uniform  cross  section  is  equal  to  the  volume  of  the 
bar  divided  by  its  length,  and  the  volume  of  the  bar  is  in  turn 
equal  to  its  mass  divided  by  the  density  of  the  material  of  which 
it  is  made.  Hence,  using  the  same  symbols  as  above,  and  in 
addition  calling  m  the  mass  of  the  conductor  and  8  its  density 
or  specific  gravity,  we  have 

A=  —  and  therefore  R=o  —  =  08  — 
8  I  ^  A     ^    m 

For  a  given  material  at  a  given  temperature  p  8  is  also  constant, 


m 

*A  mil  is  defined  as  one-thousandth  of  an  inch  and  a  circular  mil  is  de- 
fined as  the  area  of  a  circle  one  mil  in  diameter.  Since  the  area  of  a  circle 
varies  as  the  square  of  its  diameter,  the  area  of  a  circle  in  circular  mils  is 
equal  to  the  square  of  its  diameter  in  mils.  This  unit  of  area  is  therefore 
very  convenient  for  expressing  the  area  of  the  cross  section  of  a  circular  wire, 
since  the  factor  TT  is  eliminated.  The  area  of  a  circle  one-thousandth  of  an 

inch  in  diameter  is  equal  to  ^  (0.001  )2  square  inches  or  to  ^-(0.  00  1x2.  54)  2 

square  centimeters;  hence 

1  circular  mil  =  0.78540  X  10  6square  inches 

=5.0671    X  10  6square  centimeters 


134  ELECTRICAL  ENGINEERING 

where  k  is  a  constant  (equal  to  p  8)  for  a  given  material  at  a  given 
temperature.  The  specific  resistance  of  a  conductor  may  then  also 
be  expressed  indirectly  in  terms  of  this  factor  k.  When  R  is  ex- 
pressed in  ohms,  I  in  meters  and  m  in  grams,  this  factor  k  is  equal 
to  the  resistance  of  one  gram  of  the  conductor  made  into  a  wire  of 
uniform  cross  section  and  one  meter  in  length ;  this  factor  k  is  then 
said  to  be  the  specific  resistance  of  the  conductor  in  ohms  per  meter- 
gram.  It  should  be  noted  that  this  method  of  expressing  the  spe- 
cific resistance  of  a  conductor  involves  the  density  of  the  conductor, 
while  the  other  methods  do  not.  The  determination  of  the  specific 
resistance  of  a  conductor  in  meter-grams,  however,  does  not 
require  a  determination  of  the  density,  but  merely  the  measure- 
ment of  the  resistance,  length  and  mass  of  a  given  wire  of  the 

conductor,  since  k=——.     However,  to  calculate  from  the  value 

of  k  the  specific  resistance  in  microhms  per  centimeter-cube,  or 
per  inch-cube,  or  ohms  per  mil-foot,  does  require  a  knowledge  of 
the  density  of  the  conductor. 

The  various  units  of  specific  resistance  are  related  as  follows: 

1  microhm  per  inch-cube  =2.5400  microhms  per  centi- 

meter-cube 

1  microhm  per  centimeter-cube  =6.0153  ohms  per  mil-foot 

1  ohm  per  meter-gram  =  ^  microhms  per  centi- 

meter-cube 

1  ohm  per  meter-gram  __«oi.5«  o^ms  per  mji_f oo^ 

To  calculate  the  resistance  of  a  wire  of  uniform  cross  section  when 
its  specific  resistance  is  given,  the  formulas 

i  r- 

R=  p—     or     R=k — 
r  A  m 

are  directly  applicable,  but  care  must  be  employed  to  express  all 
the  quantities  entering  into  the  formula  in  the  same  system  of 
units.  For  example,  the  resistance  in  ohms  of  a  wire  which  has 
a  specific  resistance  of  1.6  microhms  per  centimeter  cube,  a 
length  of  1000  feet  and  a  cross  section  of  -J-  square  inch,  is 

^  =  1.6X10-  10QQXl2X2-54=0.0302ohms. 

0.25X(2.54)2 

Experiment  shows  that  the  resistance  of  a  given  length  of 
wire  of  uniform  cross  section  is  independent  of  the  shape  into 


CONTINUOUS   ELECTRIC  CURRENTS  135 

which  the  wire  is  bent,  provided  the  diameter  of  the  wire  is  small 
compared  to  the  radius  of  curvature  of  the  curve  into  which  it 
may  be  bent,  which  condition  is  equivalent  to  saying  that  the 
"  stream  lines  "  of  the  current  are  all  of  the  same  length  (see  Article 
101).  This  condition  is  almost  always  realized  in  practice,  and 
consequently  formulas  (11)  and  (lla)  are  in  general  directly  ap- 
plicable to  the  calculation  of  the  resistance  of  a  wire  whether  the 
wire  be  straight  or  curved  or  wound  into  a  coil  of  any  shape. 
These  formulas  are  also  applicable  to  the  calculation  of  the  resist- 
ance of  a  rod  or  bar,  provided  the  rod  or  bar  is  not  bent  into  a 
sharp  curve  and  the  distance  between  its  points  of  connection  to 
the  circuit  is  large  compared  to  the  linear  dimensions  of  its  cross 
section.  When  the  stream  lines  of  the  current  are  not  all  of  the 
same  length,  as,  for  example,  when  a  current  is  established  in  a 
heavy  short  bar  bent  into  a  sharp  curve,  the  resistance  of  the  bar 
can  be  calculated  only  when  the  distribution  of  these  stream  lines 
is  known.  Again,  when  the  stream  lines  of  the  current  are  not 
parallel,  e.g.,  the  leakage  current  through  the  insulation  of  a  cable, 
these  formulas  are  not  applicable  (see  Article  102). 

With  the  exception  of  silver,  copper  has  the  lowest  specific 
resistance  of  any  metal.  The  specific  resistances  of  a  few  common 
metals  at  0°  centigrade  are  given  below. 

Microhms  Ohms  per  Ohms  per 

per  Cm.-Cube  Mil-Foot  Meter-Gram 

Silver                                           1.49  8.94  0.156 

Copper  (very  pure,  annealed)    1.56  9.35  0.139 

Aluminum  (99%  pure)              2.56  15.4  0.067 

Iron  (very  pure)                        9.07  54.5  0.707 

Steel  rails  (average*)               14.00  84.2  1.09 

Silver  is  seldom  used  for  electric  conductors  on  account  of  its 
high  cost.  Copper  is  the  metal  most  frequently  used.  Aluminum, 
although  it  has  a  greater  resistance  than  copper  "for  the  same 
length  and  cross  section  (i.e.,  same  volume),  has,  on  account  of  its 
low  density,  a  less  resistance  for  the  same  mass  or  weight;  its 
cost  for  the  same  resistance  as  that  of  an  equal  length  of  copper 
is  about  10%  less.  It  is  therefore  used  to  a  considerable  extent 
for  overhead  transmission  lines ;  its  mechanical  strength,  however, 
is  inferior  to  that  of  copper.  Pure  iron  is  too  costly  for  use  as  a 

*The  resistance  of  steel  rails  varies  considerably  with  their  chemical 
composition. 


136  ELECTRICAL   ENGINEERING 

practical  conductor.  Steel,  however,  in  the  form  of  track  rails  or 
"  third  "  rails  is  largely  used  in  railway  work,  but  is  seldom  used 
as  a  conductor  in  the  form  of  wires,  except  for  short  telephone 
lines,  and  even  here  it  is  being  largely  supplanted  by  copper. 

Any  impurity  in  a  metal  increases  its  resistance.  All  alloys 
have  a  greater  resistance  than  that  of  the  best  conductor  in  them. 
For  certain  purposes,  particularly  in  the  construction  of  "  resist- 
ance boxes  "  and  "  rheostats,"  a  high  specific  resistance  is  de- 
sirable. Various  alloys  are  made  for  this  purpose,  the  specific 
resistances  of  which  can  be  found  in  any  electrical  engineer's 
handbook.  For  the  resistance  of  copper  and  aluminum  wires  of 
various  sizes  see  Appendix  B. 

84.  Electric  Conductance  and  Conductivity.  —  The  reciprocal 
of  electric  resistance  is  defined  as  electric  conductance.  That 
is,  when  R  is  the  resistance  of  a  conductor,  the  conductance  is 


.    . 

where  /  is  the  continuous  current  established  in  the  conductor  and 
Ph  the  rate  at  which  heat  energy  is  developed  in  it.  The  unit  of 
conductance  on  the  c.  g.  s.  electromagnetic  system  of  units  is 
one  abampere-squared  per  erg  per  second;  this-  unit  is  called  the 
absolute  unit  of  conductance  or  the  abmho.  The  practical  unit 
of  conductance  is  one  ampere-squared  per  joule  per  second,  or  one 
ampere-squared  per  watt;  this  unit  is  called  the  mho.  (Note 
that  the  word  "mho"  is  simply  the  word  ohm  written  backwards.) 

The  reciprocal  of  specific  resistance  or  resistivity  is  called 
specific  conductance  or  conductivity.  For  example,  a  specific 
resistance  of  10  ohms  per  mil-foot  is  the  same  as  a  conductivity 
of  0.1  mhos  per  mil-foot. 

85.  Matthiessen's  Standard  of  Conductivity.-  —  Matthiessen1, 
about  50  years  ago,  found  that  the  specific  resistance  of  the 
purest  copper  at  that  time  obtainable  was  0.141729  ohms  per 
meter-gram  at  0°  cent.  Matthiessen,  however,  failed  to  state  the 
density  of  his  copper,  so  that  the  exact  value  of  this  specific  re- 
sistance in  microhms  per  centimeter-cube  or  in  ohms  per  mil-foot 
is  no.t  known.  However,  assuming  8.89  as  the  probable  value 
of  this  density,  Matthiessen's  value  of  0.141729  ohms  per  meter- 
gram  is  equivalent  to  9.5900  ohms  per  mil-foot.  The  corre- 
sponding value  of  the  conductivity,  namely  0.10427  mhos  per 
mil-foot,  has  been  adopted  by  electrical  engineers  in  this  coun- 


CONTINUOUS  ELECTRIC  CURRENTS  137 

try  and  in  England  as  the  "  standard  conductivity ";  this 
standard  is  usually  referred  to  as  Matthiesseris  standard.  The 
relative  conductivity  of  any  conductor  is  then  equal  to  the  percent- 
age ratio  of  its  conductivity  per  mil-foot  at  0°  centigrade  to  the 
conductivity  (0.10427)  of  Matthiessen's  standard;  that  is,  the  rela- 
tive conductivity  of  a  conductor  which  has  a  resistance  of  15.5 
ohms  per  mil-foot  at  0°  centigrade  (i.e.,  0.0647  mhos  per  mil-foot) 
is  62%.  Note  that  the  lower  the  relative  conductivity  the  higher 
is  the  specific  resistance.  It  is  of  interest  to  note  that  copper  can 
be  made  at  present  of  a  greater  purity  than  that  used  by  Matthies- 
sen  and  consequently  having  a  relative  conductivity  greater  tha'n 
100%  of  Matthiessen's  standard.  Commercial  copper  wire  usually 
has  a  conductivity  of  from  96%  to  99% ;  the  "  harder  "  the  wire 
the  lower  its  conductivity.  Commercial  aluminum  has  a  con- 
ductivity of  about  62%. 

86.  Temperature  Coefficient  of  Electric  Resistance.  —  It  is 
found  by  experiment  that  the  variation  of  the  resistance  of  a 
conductor  with  temperature  may  be  expressed  by  the  formula 

R  =  R0(l+ftt)  (13) 

where  R0  is  the  resistance  of  the  conductor  at  any  given  "  stand- 
ard "  temperature,  R  the  resistance  of  the  conductor  at  a  tem- 
perature t  degrees  above  this  temperature,  and  ft  a  coefficient 
which  is  approximately  constant  independent  of  the  temperature 
rise  t,  but  does  depend  upon  the  temperature  corresponding  to  R0. 
Zero  degrees  centigrade  is  usually  taken  as  the  standard  tem- 
perature, and  the  value  of  this  coefficient  ft  when  the  rise  of 
temperature  is  referred  to  0°  cent,  is  called  the  resistivity  tem- 
perature coefficient  of  the  conductor,  or  briefly,  the  temperature 
coefficient  of  the  conductor.  The  value  of  ft  for  commercially 
pure  copper  depends  somewhat  upon  the  purity  of  the  metal 
and  also  upon  the  rise  of  temperature  t.  The  American  Institute 
of  Electrical  Engineers,  however,  has  adopted  the  constant 
value  0.0042  as  sufficiently  accurate  for  all  grades  of  commercially 
pure  copper  and  for  any  value  of  the  temperature  rise  ordinarily 
met  with  in  electric  machinery,  that  is,  for  any  rise  of  temperature 
not  greatly  in  excess  of  100°  cent.  The  temperature  coefficient 
of  any  other  commercially  pure  metal,  except  the  magnetic 
metals  such  as  iron,  nickel,  cobalt  and  bismuth,  has  practically 
the  same  value.  The  temperature  coefficient  for  iron  is  0.00625 
per  degree  cent. 


138  ELECTRICAL  ENGINEERING 

The  temperature  coefficient  of  carbon  is  negative  and  is  not  a 
constant.  For  example,  the  resistance  of  a  carbon-filament  lamp 
is  less  when  the  lamp  is  burning  than  when  it  is  cold.  The  tem- 
perature coefficient  of  most  insulators  is  also  negative  and  very 
large  ;  moreover,  it  is  far  from  being  constant. 

The  temperature  coefficient  of  alloys  depends  largely  upon 
their  constituents.  It  is  possible  to  make  alloys  for  which  the 
temperature  coefficient  is  zero  over  a  considerable  range  of  tem- 
perature. Such  alloys  are  extremely  useful  in  the  construction 
of  standard  resistance  coils,  since  the  resistance  of  such  a  standard 
remains  constant  for  any  ordinary  variation  of  temperature  and 
consequently  its  resistance  does  not  have  to  be  "  corrected  "  for 
temperature. 

Equation  (13)  enables  one  to  calculate  the  resistance  of  a  con- 
ductor at  any  temperature  when  its  resistance  at  any  other 
temperature  and  its  temperature  coefficient  are  known.  For, 
calling  Rt  the  resistance  of  the  conductor  at  t°  cent,  and  Rt,  its 
resistance  at  tf  degrees  centigrade,  we  have 


Hence,  taking  the  ratio  of  these  two  equations,  we  have 


Rt     1+ 


Rt,  =.  Rt  . 

L+# 

Substituting  for  ft  the  numerical  value  of  0.0042,  and  dividing 
the  numerator  and  denominator  of  the  fraction  in  the  right-hand 
member  by  0.0042,  gives 

238  +  ^ 

238  +  * 

The  operation  expressed  by  this  equation  is  readily  performed  by 
one  setting  of  a  slide  rule,  when  a  temperature  scale  is  marked 
off  on  the  lower  scale  of  the  slide,  marking  238  as  0°,  248  as  10°, 
258  as  20°,  etc.  Then,  if  the  resistance  at  25°  cent,  is  3  ohms, 
say,  the  resistance  at  any  other  temperature,  say  60°,  is  found  by 
setting  25  on  this  new  scale  opposite  3  on  the  lower  scale  of  the 
rule  and  reading  off  on  the  lower  scale  the  number  corresponding 
to  60  on  the  new  scale,  i.e.,  3.4  ohms. 


CONTINUOUS  ELECTRIC  CURRENTS  139 

Equation  (13fo)  also  gives  a  convenient  method  of  determining 
the  average  temperature  of  a  coil  of  copper  wire  when  heated  in 
any  manner,  as,  for  example,  by  the  current  established  in  it. 
For  the  resistance  of  the  coil  "  cold,"  i.e.,  at  room  temperature, 
may  be  measured  and  also  its  resistance  when  heated.  Knowing 
these  two  resistances  Rt  and  RtS  and  the  room  temperature  t,  the 
value  of  t'  is  readily  calculated.  For  example,  if  the  resistance 
of  a  coil  at  20°  cent,  is  found  to  be  5  ohms  and  its  resistance 
when  heated  6  ohms,  then  when  20  on  the  temperature  scale  of 
the  slide  rule  is  set  opposite  5  on  the  lower  scale,  the  reading  on 
the  temperature  scale  opposite  6  on  the  lower  scale  gives  the 
average  temperature,  71°  cent.  The  rise  of  temperature  due 
to  the  heating  of  the  coil  is  then  71 -20°  ==51°.  This  method 
of  measuring  the  temperature  rise  of  a  coil  is  largely  used  in 
practice. 

87.  Difference  of  Electric  Potential.—  Electric  Energy.  —  When 
a  steady  current  of  water  is  forced  through  a  pipe,  the  work  W 
done  in  any  interval  of  time  by  this  current  in  any  length  of  the 
pipe  (between  the  points  1  and  2,  say)  is  equal  to  the  drop  in  pres- 
sure V  from  1  to  2,  multiplied  by  the  quantity  of  water  Q  which 
is  forced  across  each  section  of  the  pipe  between  these  two  points 
in  this  interval,  i.e., 

W  =  VQ 

W 
or  7= — 

Q 

In  other  words,  the  drop  in  pressure  from  1  to  2  is  equal  to  the 
work  done  by,  or  the  potential  energy  lost  by,  the  current  of 
water  between  the  points  1  and  2  per  unit  quantity  of  water 
forced  through  the  pipe. 

The  drop  of  pressure  between  the  two  points  may  also  be  ex- 
pressed in  terms  of  the  quantity  of  water  per  second  forced  across 
each  section  of  the  pipe  between  these  two  points,  i.e.,  the  strength 
I  of  the  water  current,  and  the  rate  at  which  work  is  done  by  this 
current,  i.e.,  the  power  P  developed  by  the  current.  For  the 
quantity  of  water  forced  across  each  section  of  the  pipe  in  an 
interval  of  time  t  is  Q  =It  and  the  work  done  is  W  =  Pdt.  Whence, 
from  the  above  expression  for  the  drop  of  pressure 


140  ELECTRICAL  ENGINEERING 

That  is,  the  drop  of  pressure  between  any  two  points  along  a  pipe 
through  which  a  water  current  is  flowing  is  equal  to  ratio  of  the 
power  developed  by  the  current  to  the  strength  of  the  current. 
This  relation  holds  for  a  varying  current  as  well,  as  for  a  steady 
current ;  i.e.,  the  instantaneous  pressure  drop  is  equal  to  the  instan- 
taneous power  divided  by  the  instantaneous  value  of  the  current 
strength. 

When  there  is  no  pump  or  motor  connected  in  this  given 
length  of  pipe,  all  the  work  done  by  the  water  current  appears  as 
heat  energy,  that  is,  the  potential  energy  lost  by  the  water  in  this 
length  of  pipe  is  converted  into  heat  energy.  When  the  water  in 
going  from  1  to  2  passes  through  a  water-motor,  part  of  the  poten- 
tial energy  lost  by  the  water  is  converted  into  mechanical  energy 
by  the  motor.  Hence,  when  the  same  quantity  of  water  per 
second  as  before  is  forced  through  the  pipe^the  water  between  the 
points  1  and  2  will  lose  potential  energy  at  a  greater  rate,  and 
consequently  the  drop  of  pressure  between  these  points  will  be 
greater  than  before.  Again,  the  water  in  going  through  a  pump 
has  work  done  on  it,  and  consequently  the  water  in  passing 
through  the  pump  has  its  potential  energy  increased.  Hence 
the  drop  in  pressure  through  the  pump  in  the  direction  of  the 
flow  of  water  is  negative,  that  is,  there  is  a  rise  of  pressure  through 
the  pump  in  the  direction  of  the  current.  In  general,  whenever 
the  water  does  work,  or  loses  potential  energy,  there  is  a  drop  of 
pressure  in  the  direction  of  the  current,  and  whenever  work  is 
done  on  the  water,  or  the  water  gains  potential  energy,  there  is  a 
rise  of  pressure  in  the  direction  of  the  current. 

In  any  given  system  of  pipes  connecting  any  number  of  pumps 
and  motors,  in  which  the  pipes,  pumps  and  motors  are  all  com- 
pletely filled  with  water,  the  rate  at  which  work  is  done  by  the 
water  in  driving  the  motors,  in  producing  heat  energy  in  the  pipes 
and  the  water  passages  of  the  pumps  and  motors,  and  in  accelerat- 
ing itself,  is  exactly  equal  to  the  rate  at  which  work  is  done  on  the 
water  by  the  pumps.  That  is,  the  total  amount  of  potential 
energy  of  the  water  remains  unaltered ;  the  gain  in  potential  energy 
by  any  particle  of  water  as  it  is  forced  through  a  pump  is  exactly 
equal  to  the  simultaneous  loss  of  potential  energy  by  some  other 
particle  of  water  somewhere  else  in  the  system.  The  water  current 
then  acts  simply  as  a  means  for  the  transfer  of  energy ;  the  work 


CONTINUOUS  ELECTRIC  CURRENTS  141 

done  on  the  water  at  each  instant  is  exactly  equal  to  the  work 
done  by  it  at  that  instant. 

Since  the  pressure  in  a  current  of  water  has  but  a  single  value 
at  each  point  in  this  current,  it  follows  that  the  total  drop  of 
pressure  around  any  closed  path  formed  by  any  number  of  such 
pipes,  pumps  and  motors  must  be  zero.  Also,  since  the  water  is 
incompressible  (at  least,  practically  so)  the  quantity  or  volume 
of  water  flowing  up  to  any  junction  of  two  or  more  pipes  in  any 
interval  of  time  must  be  exactly  equal  to  the  quantity  of  water 
which  flows  away  from  this  junction  in  this  same  interval  of  time. 
Or,  calling  the  quantity  of  water  which  flows  away  from  any  given 
point  equivalent  to  an  equal  negative  quantity  flowing  to  that 
point,  an  equivalent  statement  of  this  same  fact  is  that  the  alge- 
braic sum  of  the  strengths  of  the  water  currents  flowing  up  to 
any  junction  of  the  pipes  is  always  zero. 

Similarly,  it  is  found  by  experiment  that  whenever  there  occur 
any  of  the  phenomena  which  are  attributed  to  the  flow  of  an 
electric  current,  there  is  always  a  loss  of  energy  by  one  or  more 
parts  of  the  circuit  or  by  some  body  in  the  vicinity  of  the  circuit, 
and  a  gain  of  energy  by  other  parts  of  the  circuit  or  by  bodies  in 
the  vicinity  of  the  circuit.  For  example,  in  the  simple  circuit 
formed  by  a  wire  connecting  the  poles  of  a  battery,  the  battery 
loses  chemical  energy,  and  the  wire  and  the  conductors  forming 
the  battery  become  heated,  or  gain  heat  energy.  The  energy  lost 
by  any  part  of  the  circuit  or  by  any  body  in  its  vicinity  is  said  to 
result  in  a  gain  of  an  equal  amount  of  electric  energy  by  the  electric 
current,  and  the  energy  gained  by  any  part  of  the  circuit  or  by 
any  body  in  its  vicinity  is  said  to  be  due  to  the  loss  of  an  equal 
amount  of  electric  energy  by  the  electric  current.  That  is,  the 
electric  energy  gained  by  the  current  in  any  portion  of  the  circuit  is 
defined  as  the  amount  of  energy  lost  by  this  part  of  the  circuit  or 
by  any  other  bodies  as  the  result  of  the  existence  of  the  current  in 
this  part  of  the  circuit;  and  the  electric  energy  lost  by  the  current 
in  any  portion  of  the  circuit  is  defined  as  the  amount  of  energy 
gained  by  this  portion  of  the  circuit  or  by  any  other  bodies  as  the 
result  of  the  existence  of  the  current  in  this  portion  of  the  circuit.* 

*Electric  energy  as  thus  defined  is  to  be  distinguished  from  electrostatic 
energy;  electric  energy  is  the  work  done  on  or  by  electric  currents,  while 
electrostatic  energy  is  energy  possessed  by  electric  charges.  Electrostatic 
energy  is  discussed  in  detail  in  Chapter  V. 


142  ELECTRICAL   ENGINEERING 

Experiment  shows  that  the  net  electric  energy,  as  thus  defined, 
gained  by  the  electric  current  in  any  closed  electric  circuit  in  any 
interval  of  time  is  zero.  (This  is  also  true  for  any  stream  line  of 
electric  current,  whether  the  current  be  continuous  or  variable.) 
That  is,  any  electric  energy  gained  by  the  current  in  any  part  of  its 
path,  e.g.,  the  energy  lost  by  a  battery  or  other  source  of  electromo- 
tive force,  is  lost  by  the  current  in  some  other  part  of  its  path, 
e.g.,  as  heat  energy,  mechanical  energy,  chemical  energy,  or  in  the 
case  of  a  variable  current,  as  magnetic  and  electrostatic  energy. 
The  electric  energy  of  an  electric  current  is  entirely  analogous  to 
the  potential  energy  of  a  current  of  water  in  a  pipe  or  system  of 
pipes  completely  filled  with  water.  The  potential  energy  gained 
by  the  water  in  any  part  of  its  path,  e.g.,  from  a  pump,  is  all  given 
out  by  the  water  in  some  other  form  of  energy,  e.g.,  the  heat  energy 
developed  in  the  pipe  and  the  work  done  on  any  hydraulic  motor 
that  may  be  connected  in  the  pipe.  An  electric  current  may  then 
be  considered  as  a  means  for  the  transfer  of  energy,  just  as  a 
stream  of  water  completely  filling  a  closed  pipe  or  system  of  pipes 
is  a  means  for  the  transfer  of  energy.  The  net  amount  of  potential 
energy  in  the  water  current  remains  constant;  similarly  the  net 
amount  of  electric  energy  in  the  electric  current  remains  constant. 

We  have  seen  how  the  rate  at  which  potential  energy  is  lost  or 
gained  by  a  current  of  water  flowing  through  a  given  length  of 
pipe  (i.e.,  the  power  developed  by  this  current)  may  be  expressed 
in  terms  of  the  strength  of  the  current  and  the  drop  of  pressure 
along  the  given  length  of  pipe.  We  are  therefore  led  to  the 
conception  of  electric  pressure,  or,  as  it  is  also  called,  electric 
potential,  as  a  property  of  an  electric  current  analogous  to  hydraulic 
pressure.  As  in  the  case  of  the  flow  of  water,  we  are  concerned  only 
with  the  difference  of  pressure  and  not  with  the  absolute  pressure  of 
the  water,  so  in  discussing  the  flow  of  electric  currents  we  need 
concern  ourselves  only  with  the  difference  of  electric  pressure 
or  difference  of  electric  potential.  We  may  then  define  the 
drop  of  electric  pressure,  or  the  drop  of  electric  potential,  from  any 
point  1  to  any  other  point  2  along  a  wire  carrying  an  electric  cur- 
rent as  the  ratio  of  the  rate  at  which  electric  energy  is  lost  by  the 
current  between  these  two  points  to  the  value  of  the  current  from 
1  to  2.  That  is,  calling  P  the  rate  at  which  electric  energy  is  lost 
by  the  current  between  the  points  1  and  2  (i.e.,  the  power  developed 
by  this  current)  and  /  the  value  of  the  current  from  1  to  2,  the 


CONTINUOUS  ELECTRIC  CURRENTS  143 

drop  of  electric  potential  from  the  point  1  to  the  point  2  is  defined 

as 

P  (14) 


which  is  a  relation  of  exactly  the  same  form  as  that  between 
hydraulic  pressure,  the  rate  at  which  potential  energy  is  lost  by 
the  current  of  water  through  the  pipe,  and  the  quantity  of  water 
per  second  flowing  through  the  pipe.  Drop  of  electric  potential, 
or  potential  difference,  is  frequently  abbreviated  "  p.d." 

The  unit  of  electric  potential  drop,  or  of  potential  difference, 
in  the  c.  g,  s.  electromagnetic  system  of  units  is  equal  to  one  erg 
per  second  per  abampere  and  is  called  the  abvolt.  When  the  rate 
at  which  energy  is  lost  by  the  current  is  expressed  in  joules  per 
second  -(i.e.,  watts)  and  the  electric  current  in  amperes,  the  unit 
of  potential  drop  is  one  joule  per  second  (i.e.,  one  watt)  per  ampere  ; 
this  unit  is  called  the  volt,  and  is  the  unit  almost  invariably  used 
in  practice.  There  is  still  another  unit  of  potential  difference 
which  is  employed  in  discussing  electrostatic  phenomena.  This 
unit  is  called  the  c.  g.  s.  electrostatic  unit.  The  relations  between 
these  various  units  are 

1  volt  =  108abvolts 

1  c.  g.  s.  electrostatic  unit     =300  volts 

1  c.  g.  s.  electrostatic  unit    =3X  1010  ab  volts 

On  the  hypothesis  that  an  electric  current  is  an  actual  flow  of 
an  incompressible  something,  the  above  definition  of  drop  of 
electric  potential  is  equivalent  to  the  definition  that  "  the  drop  of 
electric  potential  from  any  point  1  to  any  point  2  is  the  work  done 
by  unit  quantity  of  electricity  when  it  moves  from  the  point  1 
to  the  point  2."  The  definition  given  above,  however,  is  prefer- 
able, since  it  does  not  involve  any  hypothesis. 

In  designating  the  two  terminals  of  any  portion  of  a  circuit 
that  terminal  which  is  at  the  higher  potential  is  called  the  positive 
terminal  and  the  terminal  at  the  lower  potential  the  negative 
terminal. 

When  a  potential  difference  is  expressed  in  volts,  the  word 
"  voltage  "  is  commonly  employed  instead  of  the  term  potential 
difference.  The  word  "  tension  "  is  also  employed  to  mean  the 
same  thing  as  potential  difference.  For  example,  one  speaks  of 
the  voltage  of  a  generator,  a  high  tension  transformer,  etc. 


144  ELECTRICAL  ENGINEERING 

88.  Measurement  of  Drop  of  Electric  Potential.  —  Experiment 
shows  that  the  drop  of  electric  potential  as  defined  by  equation 
(14)  is  strictly  analogous  to  the  hydraulic  pressure  in  a  system 
of  pipes,  pumps  and  motors  completely  filled  with  water.  In 
particular,  it  is  found  that  the  drop  of  electric  potential  around 
any  closed  circuit  is  always  zero,  whether  this  circuit  be  a  simple 
circuit  like  that  formed  by  a  battery  with  its  poles  connected  by 
a  wire,  or  whether  this  circuit  be  part  of  any  network  of  circuits, 
no  matter  how  complicated  the  network  may  be.  Hence,  when 
a  wire  is  connected  between  any  two  points  1  and  2  of  any 
circuit  whatever,  the  drop  of  potential  from  the  point  1  to  the 
point  2  through  the  wire  will  be  exactly  equal  to  the  drop  of 
potential  from  the  point  1  to  the  point  2  through  the  conductors 
forming  the  original  circuit.  In  general,  when  a  wire  is  con- 
nected to  an  electric  circuit,  the  current  established  in  the  wire 
will  cause  a  change  in  the  strength  of  the  current  in  the  original 
circuit,  which  will  in  turn  cause  a  change  in  the  drop  of  potential 
between  the  two  points.  Hence,  in  general,  the  drop  of  potential 
in  a  wire  when  it  is  connected  to  any  two  points  of  a  circuit 
is  not  the  drop  which  originally  existed  between  these  two  points. 
However,  by  making  the  resistance  of  the  wire  sufficiently  great, 
the  current  which  is  established  in  it  can  be  made  negligibly 
small,  and  the  change  produced  in  the  current  in  the  original 
circuit  by  the  presence  of  the  wire  may  be  neglected.  Again, 
(see  Article  94)  when  any  two  dissimilar  substances  are  placed 
in  contact  a  small  difference  of  potential  is  produced  between 
them.  Hence  if  the  wire  is  made  of  a  different  material  from 
that  of  the  conductors  forming  the  circuit,  the  difference  of  po- 
tential between  the  two  ends  of  the  wire  will  not  necessarily 
be  equal  to  the  difference  of  potential  between  the  two  points  of 
the  conductor  in  contact  with  which  the  two  ends  of  the  wire 
are  placed.  This  difference,  however,  is  inappreciable  in  ordinary 
practical  measurements. 

The  fact  that  the  drop  of  electric  potential  between  any  two 
points  is  the  same  over  any  path  connecting  these  points,  leads  to 
a  simple  method  of  actually  measuring  the  potential  drop  between 
any  two  points  of  an  electric  circuit  in  which  is  established  a 
continuous  current.  For,  experiment  shows  that  when  a  continu- 
ous current  is  established  in  a  wire  which  is  of  uniform  structure 
kept  at  uniform  temperature  in  an  unvarying  magnetic  field,  then 


CONTINUOUS  ELECTRIC  CURRENTS  145 

the  only  form  of  energy  produced  in  or  around  the  wire  is  heat 
energy.  Hence  from  the  definition  of  electric  energy  given  above, 
the  total  loss  of  electric  energy  by  the  current  in  this  wire  is  equal 
to  the  heat  energy  which  is  produced  in  the  wire.  But  by  Joule's 
Law  (Article  81),  the  rate  Ph  at  which  heat  energy  is  developed 
between  any  two  points  1  and  2  of  such  a  wire  under  these  con- 
ditions, is  equal  to  RI2,  where  R  is  the  resistance  of  the  wire  be- 
tween the  points  1  and  2,  and  7  is  the  current  in  the  wire.  Hence 
the  drop  of  potential  from  the  point  1  to  the  point  2  is,  from  the 
definition  given  above  (equation  14)  , 

V=^    =  RL  (15) 


That  is,  the  drop  of  potential  between  any  two  points  1  and  2 
in  the  direction  of  the  current  in  a  wire  of  uniform  structure  and 
of  uniform  temperature  throughout,  when  this  wire  is  kept  in  an 
unvarying  magnetic  field,  is  equal  to  the  product  of  the  resistance 
of  the  wire  between  these  two  points  by  the  strength  of  the  cur- 
rent in  the  wire;  this  product  RI  is  frequently  called  the  "  resist- 
ance drop  "  between  the  two  points.  We  have  already  seen  how 
the  current  strength  and  the  resistance  may  be  measured;  conse- 
quently the  above  relation  gives  a  method  for  actually  measuring 
the  drop  of  electric  potential  between  two  points  of  such  a  wire. 

To  measure  the  drop  of  potential  between  any  two  points 
1  and  2  of  any  circuit,  it  is  then  only  necessary  to  connect 
to  these  points  a  wire  of  known  resistance  and  measure  the 
strength  of  the  current  established  in  it.  Then/if  the  resistance  of 
the  wire  is  sufficiently  high  to  make  the  current  in  the  wire  neg- 
ligibly small,  the  difference  of  potential  between  the  two  points 
is  equal  to  the  product  of  the  resistance  of  the  wire  by  the  current 
established  in  it.  This  is  the  principle  of  the  ordinary  continuous 
current  voltmeter,  which  is  simply  an  ammeter  with  a  high  resist- 
ance coil  in  series  with  it.  The  scale  of  the  instrument  is  cali- 
brated to  read  directly  in  volts  instead  of  in  amperes;  that  is, 
the  scale  reads  directly  the  product  of  the  strength  of  the  current 
through  the  wire  forming  the  coils  of  the  instrument  and  the  high 
resistance  coil  in  series  with  it  and  the  value  of  the  total  resistance 
of  this  wire.  For  accurate  measurements  with  such  an  instru- 
ment the  effect  of  changes  of  temperature  in  changing  the  resist- 
ance of  the  coils  has  to  be  taken  into  account.  Also,  care  must  be 
taken  not  to  place  the  instrument  in  a  strong  magnetic  field,  since 


146  ELECTRICAL  ENGINEERING 

such  a  field  may  alter  the  magnetic  field  in  which  the  coil  of  the 
instrument  is  designed  to  move.  The  effect  of  using  wires  of 
different  materials  in  the  various  parts  of  the  instrument,  or  for 
the  connections  to  the  instrument,  is  usually  inappreciable  in 
ordinary  practical  measurements. 

89.   Ohm's  Law.  —  Equation  (15)  may  also  be  expressed  in 
the  form 


—j 

and  in  this  form  is  known  as  Ohm's  Law,  from  the  name  of  the 
scientist  who  discovered  the  relation  expressed  by  this  formula. 
In  words,  Ohm's  Law  is  that,  when  a  continuous  current  is  estab- 
lished in  a  wire  of  uniform  structure  throughout,  kept  at  a  con- 
stant temperature  in  an  unvarying  magnetic  field,  then  the  ratio 
of  the  potential  drop  between  any  two  points  along  this  wire  to 
the  strength  of  the  current  established  in  the  wire  is  constant, 
independent  of  the  strength  of  the  current. 

90.  Electric  Power  and  Electric  Energy.  —  From  the  definition 
of  drop  of  electric  potential  given  in  Article  87,  it  follows  that  the 
rate  P  at  which  electric  energy  is  lost  by  an  electric  current  of 
strength  I  between  any  two  points  1  and  2  of  an  electric  circuit 
is  equal  to  the  drop  in  potential  V  from  the  point  1  to  the  point  2 
multiplied  by  the  value  of  the  current  from  1  to  2",  that  is 

P  =  VI  (17) 

When  an  electric  current  loses  electric  energy  it  is  said  "  to  develop 
an  amount  of  power  "  equal  to  the  rate  at  which  it  loses  energy. 
Hence  the  power  developed  by  an  electric  current  in  any  portion 
of  an  electric  circuit  is  equal  to  the  product  of  the  strength  of  the 
current  in  this  portion  of  the  circuit  by  the  drop  of  potential  in 
this  portion  of  the  circuit. 

When  the  current  I  is  measured  in  abamperes  and  the  potential 
drop  V  in  abvolts,  the  product  VI  gives  the  power  in  ergs  per 
second.  When  I  is  measured  in  amperes  and  V  in  volts  the  prod- 
uct VI  gives  the  power  in  watts.  Large  amounts  of  electric  power 
are  usually  expressed  in  kilowatts,  i.e.,  thousands  of  w^atts.  See 
Article  22  for  the  relation  between  the  various  units  of  power. 

Since  both  the  current  and  the  potential  drop  may  be  readily 
measured,  the  amount  of  power  developed  by  a  current  can  be 
readily  determined  ;  in  fact,  electric  power  can  be  measured  with 


CONTINUOUS  ELECTRIC  CURRENTS  147 

much  greater  accuracy  than  any  other  kind  of  power.  It  should 
be  noted  that  an  electric  current  can  develop  power  only  as  the 
result  of  a  loss  of  an  exactly  equal  amount  of  power  by  some 
device  associated  with  the  electric  circuit ;  that  is,  there  must 
always  be  a  generator  of  electric  power  somewhere  in  the  circuit, 
e.g.,  a  battery  or  a  dynamo  driven  by  some  external  source  of 
energy. 

The  relation  expressed  by  equation  (17)  holds  not  only  for  a 
continuous  current  but  also  for  the  instantaneous  values  of  the 
quantities  involved,  irrespective  of  how  these  quantities  may  vary 
with  time. 

The  amount  of  work  done  by  the  current  in  any  portion  of  a 
circuit  in  which  the  potential  drop  in' the  direction  of  the  current 
is  V,  when  the  current  is  continuous  for  a  time  t  and  has  a  strength 
7,  is 

W  =  VIt  (18) 

When  the  potential  drop  and  the  current  vary  with  time,  the 
total  amount  of  work  done  by  the  current  in  the  time  t  is 

(18a) 

When  the  voltage,  current  and  time  are  expressed  in  ab volts, 
abamperes  and  seconds  respectively  equations  (18)  give  the  energy 
in  ergs ;  when  these  quantities  are  expressed  in  volts,  amperes, 
and  seconds  respectively  these  formulas  give  the  energy  in  joules 
or  watt-seconds;  when  these  quantities  are  expressed  in  volts, 
amperes  and  hours  respectively,  these  formulas  give  the  energy  in 
watt-hours.  Large  amounts  of  energy  are  expressed  in  kilowatt- 
hours,  i.e.,  thousands  of  watt-hours.  For  the  relation  between 
the  various  units  of  energy  see  Article  21. 

91.  The  Wattmeter.  —  Instead  of  measuring  the  current  and 
the  potential  drop  separately  by  two  different  instruments,  it  is 
possible  to  measure  the  value  of  the  product  VI  directly  by  means 
of  a  single  instrument.  Such  an  instrument  is  called  an  electro- 
dynamometer  or  wattmeter.  This  instrument  consists  essentially  of 
two  coils,  one  of  which  is  stationary  and  the  other  mounted  inside 
the  fixed  coil  on  a  suitable  suspension  or  pivot  in  such  a  manner 
that  the  planes  of  the  two  coils  are  vertical  and  the  movable  coil 
can  turn  about  a  vertical  axis.  One  coil  Cv,  called  the  "  voltage 
coil,"  is  connected  in  parallel  with,  or  "  shunted "  across,  the 
terminals  1  and  2  of  the  circuit  in  which  it  is  desired  to 


148  ELECTRICAL  ENGINEERING 

measure  the  power,  and  the  other  coil  Cif  called  the  "  current 
coil,"  is  connected  in  series  with  the  circuit.  The  current  coil 
is  usually  stationary  and  the  voltage  coil  is  movable.  A  high 
resistance  R  is  connected  in  .series  with  the  voltage  coil,  so  that 
only  a  very  small  current  is  established  through  this  coil;  this 
current  then  depends  only  upon  the  difference  of  potential  be- 
tween the  points  1  and  2.  The  current  in  the  current  coil,  which 
is  the  same  as  the  current  in  the  line,  sets  up  a  magnetic  field 
the  strength  of  which  is  proportional  to  the  strength  of  this  cur- 
rent (see  Article  67) ;  this  magnetic  field  produces  a  moment  or 


Fig.  56. 

torque  on  the  voltage  coil,  which  torque  is  proportional  to  the 
strength  of  this  magnetic  field  and  to  the  strength  of  the  current 
established  in  the  voltage  coil.  Hence  the  torque  produced  on 
the  movable  coil  is  proportional  to  the  product  of  the  difference 
of  potential  from  1  to  2  and  the  current  in  the  circuit  between 
these  two  points.  If  the  opposing  torque  produced  by  the  sus- 
pension of  the  movable  coil,  or  by  a  suitable  spring  attached  to  it, 
is  proportional  to  the  angular  twist  given  the  coil,  its  deflection 
will  then  be  proportional  to  the  power  transferred  to  the  circuit 
between  these  two  points.  A  suitable  pointer  attached  to  the 
movable  coil  and  arranged  to  move  over  a  suitably  graduated 
scale  will  then  indicate  directly  the  value  of  the  power. 

92.  Electromotive  Force.  —  The  heat  energy  developed  in  an 
electric  circuit  due  to  the  resistance  of  the  conductors  forming  the 
circuit,  that  is,  the  heat  energy  expressed  quantitatively  by  the 
formula  Wh  =  HPt,  is  only  one  of  a  number  of  forms  of  energy  into 
which  electric  energy  is  converted.  For  example,  electric  energy 
may  manifest  itself  by  producing  chemical  changes,  that  is,  by 
being  converted  into  chemical  energy;  or  by  producing  a  mag- 


CONTINUOUS  ELECTRIC  CURRENTS  149 

netic  field,  which  in  turn  can  produce  mechanical  motion,  that  is, 
electric  energy  may  be  converted  into  magnetic  energy  which  in 
turn  is  converted  into  mechanical  energy,  etc.  Again,  whenever  an 
electric  current  is  established  some  body  or  bodies  lose  some  form 
of  energy,  which,  in  accordance  with  the  definition  of  electric 
energy,  is  converted  into  electric  energy.  For  example,  when  a 
wire  is  connected  to  the  two  poles  of  a  battery,  chemical  changes 
take  place  in  the  battery  which  result  in  a  loss  of  chemical  energy 
by  the  battery;  that  is,  the  chemical  energy  of  the  battery  is  con- 
verted into  electric  energy.  Again,  it  is  found  by  experiment 
that  work  is  required  to  change  the  number  of  lines  of  magnetic 
induction  linking  any  part  of  an  electric  circuit;  the  work  which 
is  thus  done  is,  by  the  definition  of  electric  energy,  converted  into 
electric  energy. 

In  any  portion  of  a  circuit  in  which  work  is  done  on  an  electric 
current  or  in  which  the  current  does  work,  other  than  that  done 
as  a  consequence  of  the  resistance  of  this  portion  of  the  circuit, 
there  is  said  to  exist  an  electromotive  force.  An  electromotive  force 
may  then  be  looked  upon  as  that  which  produces  or  opposes  the 
flow  of  electricity,  other  than  the  opposition  due  to  the  resistance 
of  the  conductor  in  which  it  flows.  As  the  measure  of  the  electro- 
motive force  in  any  portion  of  a  circuit  is  taken  the  rise  of  poten- 
tial which  it  would  produce  in  the  direction  of  the  current  in  this 
portion  of  the  circuit  were  there  no  resistance  drop  in  this  portion 
of  the  circuit.  As  noted  in  Article  88,  the  resistance  drop  is 
always  in  the  direction  of  the  current ;  hence  in  any  portion  of  a 
circuit  in  which  there  is  a  resistance  drop  the  electromotive  force 
is  equal  to  the  resultant  rise  of  potential  in  the  direction  of  the 
current  in  this  portion  of  the  circuit  plus  algebraically  the  resistance 
drop  in  this  portion  of  the  circuit.  An  electromotive  force  is  then 
considered  positive  with  respect  to  the  current  when  it  produces  a 
rise  of  potential  in  the  same  direction  as  that  of  the  current ; 
negative  when  it  produces  a  rise  of  potential  in  the  opposite  direc- 
tion to  that  of  the  current.  An  electromotive  force  in  the  opposite 
direction  to  that  of  the  current  is  frequently  called  a  back  or 
counter  electromotive  force.  From  the  definition  of  potential  drop 
(Article  87)  it  follows  that  wherever  the  current  and  the  electro- 
motive force  are  in  the  same  direction  there  is  a  gain  of  electric 
energy  by  the  current,  or  the  current  has  work  done  on  it ;  wher- 
ever there  is  a  back  electromotive  force  the  current  loses  electric 


150  ELECTRICAL  ENGINEERING 

energy,  or  does  work.  Electromotive  force  is  frequently  abbre- 
viated e.  m.  /. 

An  electromotive  force  is  analogous  to  the  pressure  developed 
by  a  pump,  while  a  back  electromotive  force  is  analogous  to 
the  back  pressure  due  to  hydrostatic  head  or  to  the  back 
pressure  due  to  an  hydraulic  motor.  The  resistance  drop  in  a 
conductor  is  analogous  to  the  drop  of  pressure,  or  "  loss  of  head," 
due  to  the  friction  of  a  pipe.  When  the  terminals  of  any  device 
which  develops  an  electromotive  force  are  connected  by  a  con- 
ductor, the  electromotive  force  may  be  looked  upon  as  the  cause 
of  the  flow  of  the  electric  current,  just  as  the  pressure  developed 
by  a  pump  may  be  looked  upon  as  the  cause  of  the  flow  of  the 
water  current  through  a  pipe  connecting  its  outlet  and  intake. 
Just  as  the  pressure  developed  by  the  pump  is  equal  to  the  drop 
in  pressure  in  the  pipe  and  the  water  passages  of  the  pump  due  to 
their  frictional  resistance  (provided  there  is  no  acceleration  of  the 
water  or  other  source  of  back  pressure  in  the  pipe),  so  is  the  electro- 
motive force  developed  by  any  device  equal  to  the  drop  of  electric 
pressure,  or  potential  drop,  due  to  the  electric  resistance  of  the 
conductors  forming  the  electric  circuit  (provided  the  electric  cur- 
rent is  continuous,  i.e.,  does  not  vary  with  time,  and  there  is  no 
back  electromotive  force  in  the  circuit).  Again,  when  the  outlet 
and  intake  of  a  pump  are  connected  respectively  to  the  intake 
and  outlet  of  an  hydraulic  motor,  the  pressure  developed  by  the 
pump  is  no  longer  equal  to  the  drop  in  pressure  due  to  the  fric- 
tional resistance  of  the  pipe  and  water  passages  of  the  pump  and 
motor,  but  is  equal  to  this  drop  plus  the  back  pressure  due  to  the 
motor ;  similarly,  when  any  device  which  develops  an  electromotive 
force  is  connected  by  conductors  to  another  device  which  develops 
a  back  electromotive  force  (e.g.,  an  electric  motor)  the  electromotive 
force  of  the  first  device  is  not  equal  to  the  resistance  drop  in  this 
circuit,  but  is  equal  to  this  drop  plus  the  back  electromotive  force 
developed  by  the  second  device. 

Since  an  electromotive  force  is  measured  by  the  difference  of 
potential  it  produces,  the  units  of  electromotive  force  are  the  same 
as  those  of  potential  difference,  namely,  the  abvolt,  volt,  and  the 
c.  g.  s.  electrostatic  unit  (see  Article  87). 

93.  Generalized  Ohm's  Law.  — The  relation  between  current 
strength,  electromotive  force,  potential  drop,  and  resistance  in  any 
portion  of  an  electric  circuit  in  which  a  continuous  electric  current 


CONTINUOUS   ELECTRIC  CURRENTS 


151 


is  established  may  be  stated  in  a  comprehensive  manner  in  a 
single  formula.  Fig.  57  represents  diagrammatically  any  portion 
of  an  electric  circuit;  the  terminals  of  this  circuit  are  designated 
by  the  numbers  1  and  2.  An  electromotive  force  developed  in 
this  circuit  in  the  direction  from  1  to  2  is  designated  by  the  symbol 
E12  and  an  electromotive  force  in  the  opposite  direction  by  the 
symbol  E2l.  Similarly,  a  current  in  the  direction  from  1  to  2  is 
designated  by  /12  and  a  current  in  the 
.opposite  direction  by  721.  The  drop  of 
potential  from  1  to  2  is  designated  by 
V12  and  a  drop  of  potential  from  2  to  1 
is  designated  by  F21.  The  resistance 
of  the  conductors  forming  this  part  of 
the  circuit  to  the  current  from  one  ter- 
minal to  the  other  is  represented  by  the 
symbol  R;  this  resistance  is,  of  course, 
independent  of  the  direction  of  the  circuit.  Then,  since  E12  repre- 
sents a  rise  of  potential  in  the  direction  from  1  to  2,  and  RI12  is  a 
drop  of  potential  from  1  to  2,  the  net  rise  of  potential  from  1  to  2 
is  E12—  RI12.  But  the  net  rise  of  potential  from  1  to  2  is  equal  to 
the  net  drop  of  potential  from  2  to  1 ;  hence 


or 


V2l  =  E12  —  RI12 
7    _E»-  V2l 
R 


(18a) 


The  net  rise  of  potential  from  one  terminal  to  another  of  any  por- 
tion of  an  electric  circuit  is  frequently  called  the  terminal  electro- 
motive force  of  this  portion  of  the  circuit,  or  the  electromotive 
force  impressed  upon  this  portion  of  the  circuit  by  the  rest  of  the 
circuit.  Hence,  calling  E'  12  the  terminal  or  impressed  electromo- 
tive force  in  the  direction  from  1  to  2,  we  have  that  E'  12=  —  T/r12  = 
V2l,  and  therefore  equation  (18a)  may  be  written 

E12-E'12  (186) 


/a- 


R 


Equations  (18a)  and  (186)  are  two  ways  of  expressing  the  same 
fact;  they  hold,  of  course,  only  when  the  current  entering  one 
terminal  is  the  same  as  the  current  leaving  the  other. 

Equation  (186)  is  frequently  called  the  Generalized  Ohm's  Law, 
since  this  expression  reduces  to  the  same  form  as  the  expression  for 
Ohm's  Law  (equation  16)  when  no  electromotive  force  is  developed 


152  ELECTRICAL  ENGINEERING 

T?f          V 

in  the  circuit,  i.e.,  when  E12=0  then  721  = — -  =  — .  Equation (186) 

R        R 

is  an  extremely  useful  one,  since  it  gives  a  simple  and  entirely  general 
formula  for  calculating  the  current  in  any  portion  of  a  circuit  when 
the  electromotive  force  developed  in  this  portion  of  the  circuit,  the 
impressed  or  terminal  electromotive  force,  and  the  resistance  of 
this  portion  of  the  circuit  are  known.  In  applying  this  formula 
care  must  be  taken  in  not  confusing  the  electromotive  force  de- 
veloped in  the  given  portion  of  the  circuit  with  the  electromotive 
force  impressed  across  its  terminals. 

When  the  generated  electromotive  force  E  and  the  current  / 
are  in  the  same  direction,  i.e.,  in  an  electric  generator  or  in  any 
other  device  in  which  electric  energy  is  generated,  equation  (186) 
may  be  written 

E'=E-RI  (18c) 

In  this  case  the  terminal  electromotive  Ef  force  is  always  less  than 
the  generated  electromotive  force  E.  When  the  generated  elec- 
tromotive force  E  and  the  current  /  are  in  opposite  directions,  i.e., 
in  an  electric  motor  or  in  any  other  device  which  absorbs  electric 
energy,  equation  (186)  may  be  written 

E'=E+RI  (18rf) 

In  this  case  the  impressed  electromotive  force  E'  is  greater  than 
the  back  electromotive  force  E. 

For  example,  a  continuous  current  dynamo  may  be  used 
either  as  a  generator  or  a  motor.  If  the  electromotive  force 
generated  by  the  machine  is  110  volts  in  each  case,  and  the  resist- 
ance of  its  armature  is  1  ohm,  then  when  a  current  of  10  amperes 
is  supplied  by  the  dynamo  running  as  a  generator  the  terminal 
electromotive  force  is  110 -(IX 10)  =100  volts;  while  if  the  dy- 
namo is  running  as  a  motor  the  impressed  electromotive  force 
must  be  110  +  (1X10)  =120  volts. 

In  employing  the  special  forms  (18c)  and  (ISd)  of  the  general 
equation  (186),  the  relative  directions  of  the  various  quantities 
must  not  be  lost  sight  of.  In  the  generator  equation  (18c)  the 
generated  electromotive  force  E,  the  terminal  electromotive  force 
E'  and  the  current  7  are  all  in  the  same  direction.  In  the  motor 
equation  (18d)  the  generated  electromotive  force  E  and  the  cur- 
rent /  are  in  opposite  directions;  the  impressed  electromotive 
force  Ef ,  considered  as  localized  in  the  rest  of  the  circuit  connecting 
the  terminals  of  the  portion  of  the  circuit  under  consideration,  acts 


CONTINUOUS  ELECTRIC  CURRENTS  153 

around  the  closed  circuit  of.  which  the  given  portion  is  a  part  in 
the  same  direction  as  the  current,  and  therefore  in  the  opposite 
direction  around  the  circuit  to  the  back  electromotive  force  E. 

The  student  should  bear  in  mind  that  the  various  formulas 
given  in  this  article  are  all  simply  different  ways  of  expressing  a 
single  relation.  This  relation  is  that  the  current  in  any  portion  of 
a  circuit  formed  by  one  or  more  conductors  in  series  is  equal  to  the 
algebraic  difference  between  the  generated  and  impressed  electro- 
motive forces  divided  by  the  resistance  of  the  given  portion  of  the 
circuit. 

94.  Contact  Electromotive  Force.  —  It  is  found  by  experiment 
that  whenever  an  electric  current  is  established  in  two  or  more 
conductors  connected  in  series,  the  current  always  gains  or  loses 
electric  energy  at  each  point  of  contact  between  dissimilar  con- 
ductors. For  any  given  point  of  contact,  it  is  found  that  whether 
there  is  a  gain  or  loss  of  energy  depends  upon  the  direction  of 
the  current  with  respect  to  the  two  conductors.  For  example, 
when  a  copper  wire  is  connected  to  an  iron  wire  and  a  current 
is  established  across  the  junction  of  these  two  conductors,  the 
wires  in  the  vicinity  of  the  junction  become  cooled  when  the  cur- 
rent is  in  the  direction  from  the  copper  to  the  iron,  while  if  the 
current  is  in  the  opposite  direction,  from  the  iron  to  the  copper, 
the  wires  in  the  vicinity  of  the  junction  become  heated.  These 
effects  as  a  rule  are  scarcely  appreciable,  but  when  special  pre- 
cautions are  taken  they  can  readily  be  detected.  In  the  first 
case,  then,  the  current  gains  electric  energy  at  the  junction,  and 
in  the  second  case  it  loses  electric  energy  at  the  junction. 

In  general,  then,  at  the  junction  between  any  two  dissimilar 
conductors  there  is  an  electromotive  force,  and  the  direction  of 
this  electromotive  force  is  independent  of  the  direction  of  the 
current.  Experiment  also  shows  that  the  value  of  this  electro- 
motive force  of  contact,  as  it  is  called,  does  not  depend  upon  the 
strength  of  the  current  or  upon  the  area  or  shape  of  the  surface 
of  contact  between  the  two  conductors,  but  depends  only  upon 
the  nature  of  the  two  conductors  in  contact  and  upon  the  tem- 
perature of  the  junction.  The  value  of  this  contact  electromotive 
force  between  metallic  conductors  is  quite  small,  only  a  small 
fraction  of  a  volt  (e.g.,  between  copper  and  zinc  at  25  degrees 
centigrade  it  is  0.00045  volt),  and  in  practical  work  it  is  therefore 
usually  negligible.  However,  in  case  of  a  metal  conductor  in 


154  ELECTRICAL  ENGINEERING 

contact  with  an  electrolyte  the  contact  electromotive  force  may 
be  several  volts.  In  this  case,  only  a  very  small  part  of  the 
electric  energy  absorbed  by  or  given  out  by  the  current  manifests 
itself  as  a  loss  or  gain  of  heat  energy  at  the  junction  of  the  two 
conductors,  but  the  gain  or  loss  of  electric  energy  by  the  current 
appears  as  a  loss  or  gain  of  chemical  energy  at  the  junction  of 
the  two  conductors.  That  is,  the  transfer  of  energy  involved 
in  the  chemical  changes  which  take  place  at  the  junction  is  many 
times  greater  than  the  transfer  of  heat  energy  from  or  to  the 
junction  between  dissimilar  metals.  As  in  the  case  of  two  metals 
in  contact,  however,  this  contact  electromotive  force  between 
conductors  and  electrolytes  is  independent  of  the  strength  or  the 
direction  of  the  current  through  the  junction  and  is  also  inde- 
pendent of  the  area  and  the  shape  of  the  surface  of  contact,  but 
depends  only  upon  the  ^nature  of  the  conductors  in  contact  and 
the  temperature  of  the  conductors  at  the  junction. 

It  is  found  by  experiment  that  whenever  any  number  of 
conductors  are  connected  in  series,  and  there  are  no  electromotive 
forces  in  this  chain  of  conductors  other  than  the  electromotive 
forces  of  contact  at  their  junctions,  then  the  net  electromotive 
force  between  the  ends  of  this  series  of  conductors  is  the  same 
as  if  the  two  end  conductors  were  connected  directly  to  each 
other,  provided  all  the  conductors  are  kept  at  the  same  uniform 
temperature  and  the  chemical  action  at  the  cathode  in  each  electrolyte 
in  the  series  is  just  the  reverse  of  the  chemical  action  at  the  anode 
in  this  electrolyte.  This  fact  is  sometimes  called  the  Law  of  Succes- 
sive Contacts.  For  example,  if  a  copper  wire  is  connected  to 
an  iron  wire  which  in  turn  is  connected  to  an  aluminum  wire, 
the  net  electromotive  force  between  the  free  ends  of  the  copper 
and  the  aluminum  wire  is  the  same  as  if  the  copper  wire  were 
connected  directly  to  the  aluminum  wire.  Similarly,  when  two 
copper  wires  are  soldered  together,  the  net  effect  of  the  solder  is 
nil,  whether  the  two  copper  wires  are  in  actual  contact  or  are 
joined  only  through  the  solder.  (If  the  flux  used  in  soldering  acts 
chemically  on  the  wires,  an  electromotive  force  may  be  produced 
at  the  junction ;  this  electromotive  force,  although  small,  may  be 
sufficiently  large  to  cause  considerable  trouble  when  delicate  meas- 
urements are  to  be  made.)  Again,  the  net  electromotive  force  of 
a  silver  voltameter  is  zero,  since  at  the  anode  silver  goes  into  solu- 
tion as  silver  nitrate  and  at  the  cathode  silver  is  deposited  from 


CONTINUOUS  ELECTRIC  CURRENTS  155 

the  silver  nitrate  solution,  that  is,  the  chemical  actions  at  the  two 
electrodes  are  just  the  reverse  of  each  other. 

The  two  exceptions  to  this  law  of  successive  contacts  are 

1.  When  the  chemical  actions  at  the  two  electrodes  in  any 
electrolyte  in  the  series  are  not  the  reverse  of  each  other,  and 

2.  When  the  conductors  forming  the  series  are  not  at  the  same 
temperature. 

In  either  case,  when  the  conductors  forming  the  ends  of  the 
series  are  connected  together,  there  will  in  general  be  a  current 
established  in  this  closed  chain,  since  the  net  rise  of  potential 
due  to  the  net  electromotive  force  in  the  series  must  be  balanced 
by  an  equal  fall  of  potential,  for  the  total  drop  of  potential 
around  any  closed  circuit  is  always  zero.  The  value  of  the 
current  established  in  the  closed  circuit  thus  formed  will  be 
equal  to  the  resultant  or  net  electromotive  force  divided  by 
the  total  resistance  of  all  the  conductors  in  the  series  (see  Arti- 
cle 98). 

95.  Chemical  Batteries.  —  The  action  of  all  chemical  batteries 
is  based  upon  the  first  exception  to  the  Law  of  Successive  Con- 
tacts. For  example,  in  the  case  of  the  simple  copper-sulphuric 
acid-zinc  battery  described  at  the  beginning  of  this  chapter,  the 
electromotive  force  of  contact  between  the  zinc  and  the  sulphuric 
acid  solution  is  in  the  direction  from  the  zinc  to  the  acid  and 
is  about  one  volt  greater  than  the  contact  electromotive  force 
between  the  copper  and  the  acid.  This  latter  electromotive 
force  is  in  the  direction  from  the  copper  to  the  acid;  hence 
the  net  electromotive  force  of  the  battery  from  the  zinc  pole 
to  the  copper  pole  is  about  one  volt.  The  copper  pole  is  the  posi- 
tive pole  and  the  zinc  pole  the  negative  pole.  *  That  is,  there  is  a 
net  rise  of  potential  through  the  battery  from  the  zinc  to  the  cop- 
per pole  equal  in  value  to  about  one  volt.  Consequently,  when 
the  two  poles  of  the  battery  are  connected  by  a  copper  wire, 
a  current  is  established  in  this  wire  equal  in  value  to  this 
electromotive  force  divided  by  the  total  resistance  of  all  the 
conductors  in  series.  For  example,  calling  the  electromotive 
force  of  the  battery  1  volt,  the  resistance  of  the  wire  1  ohm, 
*The  standard  symbol  for  a  battery  is  two  parallel  lines  thus 

I 

The  short  line  represents  the  negative  and  the  long  line  the  positive  pole. 


156  ELECTRICAL  ENGINEERING 

and  the  resistance  of  the  conductors  forming  the  battery  0.2  ohms, 
the  current  in  the  circuit  will  be  —  =0.833  amperes.  The  resistance 

i.Zi 

of  the  conductors  forming  the  battery  is  usually  called  the  internal 
resistance  of  the  battery,  and  is  never  negligible  unless  the  wire 
forming  the  external  circuit  has  a  very  large  resistance.  It 
should  also  be  noted  that  in  the  example  just  cited  there  is  also 
a  contact  electromotive  force  between  the  zinc  pole  and  the  copper 
wire  forming  the  external  circuit,  but  since  this  electromotive 
force  is  but  a  very  small  fraction  of  a  volt  it  is  usually  negligible 
in  practical  work.  For  a  description  of  the  construction  of  the 
common  forms  of  chemical  batteries  used  in  practice,  see  any  elec- 
trical engineer's  handbook. 

It  is  found  by  experiment  that  the  net  electromotive  force 
of  any  chemical  battery  or  cell  is  constant,  provided  the  chemical 
nature  of  the  electrodes  and  the  electrolyte  in  contact  with  them 
does  not  change.  One  electrode  may  waste  away  and  the  other 
increase  in  mass,  due  to  the  chemical  action  which  takes  place 
at  them,  but  as  long  as  their  chemical  composition  and  that  of 
the  electrolyte  in  contact  with  them  does  not  change  and  the 
temperature  remains  constant,  the  electromotive  force  of  the 
battery  remains  constant.  In  general,  however,  when  an  electric 
current  is  established  through  a  battery,  the  electrodes  not  only 
change  in  mass,  but  some  of  the  products  of  the  chemical  actions 
which  take  place  collect  at  the  electrodes,  and  thus  change  the 
nature  of  the  substances  in  contact,  and  consequently  the 
electromotive  force  of  the  battery  changes.  For  example,  when 
a  current  is  established  through  the  copper-sulphuric-acid-zinc 
battery,  zinc  sulphate  is  formed  at  the  zinc  electrode,  and 
bubbles  of  hydrogen  gas  collect  at  the  copper  electrode  and 
the  electromotive  force  of  the  battery  falls  off.  The  battery  in 
such  a  case  is  said  to  become  polarised,  and  the  decrease  of  its 
electromotive  force  is  said  to  arise  from  a  back  electromotive  force 
of  polarisation.  There  are  various  methods  for  preventing  the 
polarisation  of  a  battery;  see  any  electrical  engineer's  handbook. 
The  polarisation  of  a  dry  battery  is  particularly  noticeable,  since 
the  electrolyte  in  such  a  battery,  instead  of  filling  the  entire 
space  between  the  poles  of  the  battery,  simply  impregnates  a 
practically  solid  mass  between  these  poles.  Consequently,  the 
products  of  the  chemical  actions  at  the  poles  of  the  battery  cannot 


CONTINUOUS   ELECTRIC  CURRENTS  157 

diffuse  rapidly  through  the  battery  but  collect  at  the  poles  where 
they  are  formed.  However,  when  such  a  battery  is  left  open- 
circuited  for  a  time  after  it  is  used,  these  products  gradually  dif- 
fuse through  the  battery,  and  its  electromotive  force  gradually 
returns  to  practically  its  original  value,  provided  none  of  the 
active  materials  has  been  completely  destroyed. 

96.  Definition    of    the    International    Volt.  —  Many    of    the 
ordinary  forms  of  batteries  undergo  chemical  changes  in  their 
various  parts  even  when  left  open-circuited.     This  is  due  chiefly 

.  to  impurities  in  the  chemicals  of  which  they  are  made.  It  is 
possible,  however,  to  construct  a  battery  which  will  remain 
practically  unaltered  for  several  years,  provided  no  current  is 
taken  from  it.  Hence  such  a  battery,  or  cell,  makes  a  very 
convenient  standard  of  electromotive  force  or  potential  difference. 
One  of  the  most  satisfactory  cells  of  this  kind  is  that  known  as 
the  Clark  cell,  and  in  terms  of  this  cell  the  International  Congress 
of  Electricians  defined  the  volt  as  follows : 

"  As  a  unit  of  electromotive  force  (shall  be  taken)  the  Inter- 
national Volt,  which  is  the  E.M.F.  that  steadily  applied  to  a 
conductor  whose  resistance  is  one  international  ohm,  will  produce 
a  current  of  one  international  ampere,  and  which  is  represented 
sufficiently  well  for  practical  use  by  Hii  of  the  E.M.F.  between 
the  poles  or  electrodes  of  the  voltaic  cell  known  as  Clark's  cell, 
at  a  temperature  of  15°  C.,  and  prepared  in  the  manner  described 
in  the  accompanying  specification  (B)." 

The  specification  referred  to  is  to  be  found  on  page  10  of 
Foster's  Electrical  Engineer's  Pocket  Book.  This  definition  has 
been  legalized  by  most  civilized  countries. 

In  the  practical  use  of  a  Clark  cell  as  a  standard  of  comparison 
of  potential  differences,  an  arrangement  is  used  which  obviates 
the  necessity  of  taking  any  current  from  the  cell.  The  principle 
of  this  method  is  to  balance  the  electromotive  force  of  the  cell 
against  an  equal  resistance  drop.  See  Article  100. 

97.  Thermal  Electromotive  Forces.  —  As  noted  in  Article  94 
the  Law  of  Successive  Contacts  does  not  hold  in  case  the  tempera- 
ture of  the  chain  of  conductors  forming  the  electric  circuit  is 
not  the  same  for  all  these  conductors.     For  example,  when  an 
iron  and  a  copper  wire  are  connected  to  each  other  at  their  two 
ends  in  such  a  manner  that  they  form  a  closed  loop,  the  electro- 
motive force,  and  therefore  the  current,  in  this  loop  will  not  be 


158  ELECTRICAL  ENGINEERING 

zero  if  the  two  junctions  between  the  iron  and  the  copper  are 
kept  at  different  temperatures,  or  even  if  there  is  a  difference 
of  temperature  between  any  two  points  of  the  same  wire.  These 
thermal  electromotive  forces,  as  they  are  called,  are  however  only 
a  small  fraction  of  a  volt  even  for  a  considerable  difference  of 
temperature,  and  consequently  in  any  circuit  in  which  there 
are  other  electromotive  forces  of  the  order  of  a  volt  or  more, 
they  may  be  neglected.  One  important  practical  application 
of  these  thermal  electromotive  forces,  however,  is  in  the  thermo- 
electric couple  or  electric  pyrometer  for  measuring  high  tempera- 
tures. The  ordinary  form  of  electric  pyrometer  consists  essen- 
tially of  a  platinum  and  a  platinum-iridium  wire  fused  together 
at  one  end  and  connected  in  series  with  a  millivoltmeter  (i.e.,  a. 
voltmeter  designed  to  measure  differences  of  potential  of  the 
order  of  a  thousandth  of  a  volt).  The  junction  between  the  two 
wires,  suitably  protected  by  a  porcelain  or  quartz  tube,  is  placed 
in  the  furnace  the  temperature  of  which  it  is  desired  to  measure. 
The  difference  of  potential  indicated  by  the  voltmeter  is  then 
practically  proportional  to  the  difference  between  the  temperature 
of  the  hot  junction  in  the  furnace  and  the  temperature  of  the 
ends  of  the  two  wires  where  they  are  connected  to  the  voltmeter. 
98.  Kirchhoff's  Laws.  —  We  have  already  had  occasion,  in 
several  instances,  to  make  use  of  the  two  experimental  facts  that 

1.  The  algebraic  sum  of  the  currents  coming  up  to  any  junction 
in  a  network  of  conductors  is  always  zero,  and 

2.  The   algebraic   sum   of   the  potential   drops   around   any 
closed  loop  in  a  network  of  conductors  is  always  zero. 

These  two  experimental  facts  are  known  as  Kirchhoff's  Laws, 
from  the  name  of  the  scientist  who  first  clearly  enunciated  them. 
By  making  use  of  these  facts  one  can  always  predetermine  (1) 
the  currents  in  each  branch  of  a  network  when  the  resistance 
of  each  branch  and  the  electromotive  force  in  each  branch  are 
known,  or  (2)  the  resistance  of  each  branch  of  a  network  when 
the  current  in  each  branch  and  the  electromotive  force  in  each 
branch  are  known,  or  (3)  the  electromotive  force  in  each  branch 
when  the  current  in  each  branch  and  the  resistance  of  each  branch 
are  known.  These  two  laws  are  therefore  of  fundamental  im- 
portance. 

It  should  be  carefully  borne  in  mind  in  applying  these  laws 
that  a  current  leaving  any  point  is  equivalent  to  an  equal  negative 


CONTINUOUS  ELECTRIC  CURRENTS  159 

current  entering  that  point,  and  that  an  e.  m.  f.  in  any  chosen 
direction  is  equivalent  to  a  rise  of  potential  in  that  direction. 
In  working  out  any  problem  concerning  a  network  of  circuits  it 
is  convenient  to  make  a  diagram  of  the  network  and  to  place  on 
each  branch  in  this  diagram  a  number  or  symbol  to  represent  the 
value  of  the  current  in  this  branch  and  an  arrow  or  subscripts 
to  indicate  the  direction  of  the  current  represented  by  this  num- 
ber or  symbol,  and  wherever  there  is  an  e.  m.  /.  to  place  a  number 
or  symbol  to  represent  the  value  of  this  e.  m.  f.  and  an  arrow  or 
subscripts  to  indicate  its  direction.  Then  at  any  junction  point 
those  currents  represented  by  arrows  pointing  toward  the  point 
are  to  be  considered  positive  (say)  and  those  represented  by 
arrows  pointing  away  from  the  point  are  to  be  considered  nega- 
tive; and  for  any  closed  loop  those  currents  and  e.  m.f's  repre- 
sented by  arrows  pointing  around  the  loop  in  the  clockwise 
direction  (say)  are  to  be  considered  positive  and  those  pointing 
around  the  loop  in  the  counter-clockwise  direction  are  to  be  con- 
sidered negative.  With  this  understanding,  we  then  have, 

S  /=0  at  every  point  (19a) 

2  E  =2  R  I  for  every  closed  loop  (196) 

where  /,  R,  and  E  represent  the  current,  the  resistance  and  the 
e.  m.  f.  respectively  in  each  branch  of  the  loop,  and  the  symbol 
2  indicates  the  algebraic  sum  of  the  expression  following  it. 

Equations  (19)  enable  one  to  write  down  a  set  of  simultaneous 
equations  for  the  given  network,  but  it  will  be  found  that  at 
least  one  of  the  current  equations  may  be  derived  directly  from 
the  other  current  equations,  and  that  at  least  one  of  the 
potential  equations  may  be  derived  from  the  other  potential 
equations.  That  is,  the  number  of  independent  equations  of 
each  form  will  be  one  less  than  the  number  which  it  is  possible 
to  write  down.  It  should  also  be  noted  that  it  is  frequently 
unnecessary  to  write  down  formally  all  the  possible  independent 
equations;  many  of  the  simpler  problems  can  be  solved  by  writ- 
ing down  two  independent  expressions  for  the  potential  drop 
between  each  pair  of  points  and  equating  these  two  expressions; 
this  is  illustrated  by  the  Generalized  Ohm's  Law  (Article  93)  which 
is  but  a  special  case  of  Kirchhoff's  second  law.  The  following 
examples  will  serve  to  indicate  the  use  of  KirchhofFs  Laws :  * 

*The  solution  of  network  problems  by  means  of  determinants  is  given 
in  detail  in  Del  Mar's  Electric  Power  Conductors. 


160  ELECTRICAL  ENGINEERING 

a.  Resistances  and  Electromotive  Forces  in  Series.  —  Con- 
sider any  number  of  conductors  in  series  (Fig.  58)  and  let  a  drop  of 
potential  F12  be  established  between  the  two  ends  of  this  series  of 
conductors.  Let  the  current  in  these  conductors  be  in  the  direc- 
tion from  1  to  2  and  let  I12  be  its  value;  I12  will  be  the  same  in  each 
conductor,  since  they  are  in  series.  Let  R'  ',  R",  Rf",  etc.,  be  the 
resistances  of  the  various  conductors  and  E'12,  E"12,  E'"12,  etc., 
the  electromotive  forces  in  this  portion  of  the  circuit  between  the 
points  1  and  2  in  the  direction  from  1  to  2.  Then  the  potential 
drop  from  1  to  2  is  also  #'/12-  E'12+R"  712-  E"»+  R'"  I12-E'"12, 
etc.  Hence 

V12=(R'  +  R"+R'",  etc.)  Ii2-(E'12+E»12+E'"12;etc.).   (20) 
Therefore  the  resistances  between  the  points  1  and  2  are  equivalent 
to  a  single  resistance 

R  =  R'  +  R»+R"'}  etc.  (20a) 

and  the  electromotive   forces  between  the  points   1   and  2  are 
equivalent  to  a  single  electromotive  force 

E12  =  E'12+  E"12+  W"12,  etc.  (206) 

When  the  equivalent  electromotive  force  E12  is  positive  and  greater 
than  the  product  of  the  current  I12  and  the  equivalent  resistance  R, 


i-A/WWH  U/WWH  U/WWH 

R'  R"  R'"  - 


'  '12 

Fig.  58. 

the  drop  of  potential  from  1  to  2  will  be  negative,  that  is,  there  will 
be  an  actual  rise  of  potential  from  1  to  2;  this  corresponds  to  the 
condition  when  the  part  of  the  circuit  from  1  to  2  is  supplying 
energy  to  the  rest  of  the  circuit  which  completes  the  closed  loop 
from  2  to  1.  When  E12  is  negative  or  is  less  than  the  product 
R  I12,  the  potential  drop  from  1  to  2  is  positive,  which  represents 
a  transfer  of  energy  to  this  part  of  the  circuit  from  the  portion 
of  the  circuit  closing  the  loop  from  2  to  1. 

b.  Resistances  and  Electromotive  Forces  in  Parallel.  —  When 
a  number  of  resistances  R',  R",  R'",  etc.,  are  connected  in  parallel 
between  two  points  1  and  2,  and  there  are  electromotive  forces 


CONTINUOUS  ELECTRIC  CURRENTS 


161 


L12 


E'J2,  E'\2,  7£'"]2,  etc.,  in  these  respective  branches,  the  drop  of 
potential  FJ2  from  the  point  1  to  the  point  2  must  be  the  same 
for  each  branch,  and  the  total  Ej 

current  entering  the  point  1 
must  be  equal  to  the  total 
current  leaving  this  point. 
Hence,  calling  712  the  total 
current  entering  the  point  1 
in  the  direction  from  1  to  2, 
and  7'12,  7"12,  7"'12,  etc.,  the 
currents  in  the  respective  Fig-  59. 

branches  in  the  direction  from  1  to  2,  we  have 
712=7'12+7"12+7'"]2  +  etc. 
and 

Rf  Ifl2-  E'12  =  R"  7"12-  E\2  =  R"'  7"'12-  #'"12=etc.         (21a) 
from  which  relations  the  currents  in  the  individual  branches  may 
be  calculated  when  the  resistances  and  the  electromotive  forces 
in  these  branches  are  known.     Again,  if  the  drop  of  potential 


(21) 


V1  from 


is  given  instead  of  the  total  current  712,  we  have 


instead  of  the  first  equation  the  relations 

V12  =  R'  f»-  E',2  =  R»  7"12-  E"»  =  R"  7"'12-#'"12          (216) 
which  are  also  sufficient  for  the  calculation  of  the  currents  in  the  in- 
dividual branches.     In  case  there  are  no  electromotive  forces  in  the 

branches  between  the  points  1  and  2,  we  have  from  equation  (216), 

y 
putting  R  =  —  -,  where  712  is  the  total  current  from  1  to  2,  that 

/12  R  I12=R'  riz  =  R"ri2=R'"  7"'12 

whence 


7/ 


1      1"        1      T 

L  1  o        L      10  •*  1  •>       •*- 


etc. 


R      Rf     R      R"    R         R"r 

therefore,  adding  these  equations  and  substituting  for  712  its  value 
from  equation  (21),  gives 


1111, 

—  = — | 1 f-etc. 

72     Rf     R"     R"f 


(21c) 


That  is,  the  equivalent  resistance  R  of  any  number  of  branch  cir- 
cuits in  parallel,  in  which  there  are  no  electromotive  forces,  is  the 
reciprocal  of  the  sum  of  the  reciprocals  of  the  resistances  of  the 

individual  branches;  or,  the  equivalent  conductance  \  =—)  of  any 

R 

number  of  branch  circuits  in  parallel,  in  which  there  are  no  electro- 


162 


ELECTRICAL   ENGINEERING 


motive  forces,  is  equal  to  the  sum  of  the  conductances  of  the  in- 
dividual branches.  It  also  follows  from  equation  (216)  that  when 
there  are  any  number  of  branch  circuits  connecting  any  two  points, 
and  there  are  no  electromotive  forces  in  these  branches,  the  total  cur- 
rent divides  among  the  various  branches  in  such  a  manner  that  the 
ratio  of  the  current  in  any  branch  to  the  current  in  any  other 
branch  is  equal  to  the  inverse  ratio  of  the  resistances  of  these  two 
branches.  It  should  be  carefully  noted  that  none  of  these  relations 
are  true  when  there  are  electromotive  forces  in  any  of  the  branches. 
In  the  special  case  of  two  resistances  in  parallel,  but  no  e.  m.  f. 
in  either  branch,  equation  (21  c)  becomes 

R'  XR"  (21d) 

99.  The  Wheatstone  Bridge.  —  A  special  'arrangement  of 
electric  circuits  which  is  extensively  used  in  the  comparison  of 
resistances  is  that  known  as  the  Wheatstone  Bridge,  and  is  shown 
diagrammatically  in  Fig.  60.  B  is  a  battery  of  any  kind.  G  a 
galvanometer,  and  Rlt  R2,  R3,  and  #4  are  the  resistances  of  the 

B 


branches  between  the  points  1  and  2,  2  and  3,  3  and  4,  and  4  and  1 
respectively.  The  currents  in  each  branch  of  this  network  can  be 
calculated  for  any  values  of  these  resistances  when  the  electromo- 
tive force  of  the  battery,  and  the  resistances  of  the  battery  and 
the  galvanometer  (including  the  connecting  leads)  are  known. 
However,  there  is  a  simple  relation  among  the  four  resistances 
Rlt  R2,  R3,  and  #4  for  which  the  current  in  the  galvanometer 
circuit  will  be  zero,  independent  of  the  electromotive  force  of  the 
battery  and  the  resistances  of  the  battery  and  galvanometer  cir- 
cuits. The  condition  that  there  be  no  current  in  the  galvanometer 
is  that  there  be  no  difference  of  potential  across  its  terminals. 


CONTINUOUS  ELECTRIC  CURRENTS  163 

Call  712,  723,  714,  and  743  the  currents  in  the  branches  12,  23,  14,  and 
43  respectively,  the  order  of  the  subscripts  indicating  the  direction 
of  the  current  in  each  instance.  Applying  Kirchhoff's  first  law  to 
the  points  2  and  4  we  then  have  for  no  current  in  the  galvanometer, 


and  applying  KirchhofFs  second  law  .to  the  loops  1241  and  2342 
we  have 

R,  Il2  =  Rt  714  and  R2  I23  =  R3  743 

but  since  712=723  and  714=743,  the  last  equation  may  be  written 
R2  112  =  R3  7j4.  Whence,  taking  the  ratio  of  this  equation  and  the 
equation  Rv  I12  =  R4  714  we  get 

R2R3 


or  „ 

R4=^-.R3  (22) 

/to 

Hence,  when  the  ratio  of  the  two  resistances  Rl  and  R2  is  known, 
and  the  resistance  R3  is  changed  until  there  is  no  current  in  the 
galvanometer,  which  will  be  indicated  by  the  galvanometer  show- 
ing no  deflection  when  connected  into  the  circuit,  the  resistance 
R4  can  be  calculated  from  this  equation. 

In  the  simplest  form  of  Wheatstone  Bridge,  the  resistances 
R!  and  R2  are  formed  by  a  continuous  wire  of  uniform  cross 
section  and  the  resistance  R3  is  formed  of  a  single  standard  re- 
sistance coil.  Instead  of  altering  this  resistance  R3,  the  galva- 
nometer terminal  2  is  moved  along  the  wire  until  the  galvanometer 
deflection  becomes  zero.  The  ratio  of  the  two  resistances  Rl 
and  R2  is  then  equal  to  the  ratio  of  the  lengths  of  this  wire  between 
the  points  12  and  23  respectively.  Calling  these  lengths  112  and 
/23  respectively,  we  then  have  that 

Rt=lf.R3  (22a) 

^23 

Whence  it  is  possible  to  determine  by  a  very  simple  experiment 
the  resistance  of  any  conductor  in  terms  of  a  single  standard 
resistance.  Hence  it  is  necessary  to  measure  absolutely  (see 
Article  82)  the  resistance  of  but  a  single  "  standard  "  of  resistance, 
and  all  other  resistances  can  be  expressed  directly  in  terms  of 
this  standard. 

100.  The  Potentiometer.  —  Another  important  network  is  that 
used  in  the  so-called  "  potentiometer  method  "  of  comparing 


164 


ELECTRICAL  ENGINEERING 


potential   differences.     A    simple    arrangement  of    this    kind   is 
shown  in  Fig.  61.     B  is   a  battery  or  other   source  of-  electro- 


Fig.  61. 


motive  force  which  is  connected  to  the  two  ends  of  a  wire  of 
uniform  cross  section.  To  the  end  1  of  this  wire  is  also  connected 
a  standard  cell  C,  say  a  Clark  cell,  in  series  with  a  galvanometer  G 
and  a  high  resistance  to  prevent  a  large  current  from  flowing 
through  the  cell  while  the  adjustments  are  being  made.  The 
other  end  of  this  circuit  is  an  adjustable  contact  3  which  can  be 
moved  along  the  wire.  The  like  poles  of  the  two  batteries  must 
be  connected  to  the  same  end  of  the  wire,  and  the  electromotive 
force  of  the  battery  B  must  be  greater  than  the  electromotive 
force  of  the  standard  cell.  Let  the  contact  3  be  moved  along 
the  wire  until  the  galvanometer  shows  no  deflection  (in  the  final 
adjustment  the  high  resistance  is  to  be  short-circuited).  Then, 
applying  KirchhofFs  second  law  to  the  loop  1C<731;  we  have, 
since  there  is  no  current  in  the  galvanometer,  that 


where  E  is  the  electromotive  force  of  the  standard  cell,  R13  the 
resistance  of  the  wire  from  1  to  3,  and  /  the  current  in  the  wire. 

Tjl 

The  potential  drop  per  unit  length  of  the  wire  will  then  be  — 

^13 

where  113  is  the  length  of  the  wire  between  1  and  3,  and  the  potential 
drop  across  any  length  of  the  wire,  llx  say,  will  then  be  —E. 

'l3 

Hence,  if  the  standard  cell  is  replaced  by  any  other  battery,  the 
electromotive  force  E'  of  which  is  to  be  measured,  and  x  is 
the  position  of  the  sliding  contact  at  which  the  galvanometer 
shows  no  deflection,  the  electromotive  force  of  this  battery  will  be 


CONTINUOUS  ELECTRIC  CURRENTS  165 

Ef  =—  E,  provided  the  current  from  the  battery  B,  and  therefore 

HI 
the  current  in  the  wire  12,  does  not  change. 

This  potentiometer  method  of  comparing  electromotive  forces 
is  extensively  employed  in  calibrating  electrical  measuring  in- 
struments. In  practice,  the  wire  12  is  usually  replaced  in 
whole  or  in  part  by  a  set  of  resistance  coils,  the  resistances  of 
which  are  accurately  known. 

101 .  Stream  Lines  of  Electric  Current.  —  So  far  we  have  con- 
fined our  attention  to  conductors  which  are  in  the  form  of  wires 
or  long  rods  or  bars,  which  we  have  assumed  may  be  considered 
as  geometrical  lines.  Certain  problems  arise  in  practice,  however, 
when  this  method  is  not  permissible;  for  example,  the  calculation 
of  the  insulation  resistance  of  a  cable,  the  calculation  of  the 
magnetic  field  intensity  within  the  substance  of  a  wire,  etc.  In 
such  cases  it  is  necessary  to  look  upon  the  conductor  carrying 
the  current  as  made  up  of  "  current  filaments,"  just  as  a  mag- 
netised body  is  conceived  to  be  made  up  of  magnetic  filaments. 

Experiment  shows  that  when  an  insulating  gap  is  cut  in  a 
conductor  in  which  an  electric  current  is  established,  this  gap  in 
general  produces  a  change  in  the  amount  of  the  effects  produced 
in  and  around  the  conductor,  e.g.,  a  change  in  the  force  produced 
on  the  conductor  by  a  magnetic  field,  a  change  in  the  amount  of 
heating  produced,  etc.  The  amount  of  these  changes  produced 
depends  upon  the  direction  in  which  the  gap  is  cut.  It  is  possible 
to  cut  a  very  narrow  gap  in  the  conductor  in  such  a  direction  that 
no  change  whatever  is  produced  in  any  of  these  effects,  just  as 
it  is  possible  to  cut  a  very  narrow  gap  in  a  magnet  without  pro- 
ducing any  new  poles  (see  Article  46).  A  conductor  of  any  shape 
whatever  in  which  a  current  is  established  may  then  be  divided 
into  filaments  separated  from  each  other  by  insulating  walls  of 
infinitesimal  thickness  without  producing  any  change  in  the  effects 
produced  in  and  around  the  conductor.  These  filaments  may  be 
considered  of  very  small  cross  section  and  may  therefore  be 
treated  as  geometrical  lines.  They  may  be  looked  upon  as  the 
lines  along  which  the  electric  current  flows;  these  filaments  are 
called  the  stream  lines  of  the  electric  current.  Such  stream  lines 
are  analogous  to  hydraulic  stream  lines. 

Since  each  of  these  current  filaments  may  be  insulated  from 
the  rest  of  the  conductor  without  altering  the  phenomena  pro- 


166  ELECTRICAL  ENGINEERING 

duced  by  the  electric  current,  each  filament  may  be  treated  as  an 
insulated  wire.  The  definitions  of  the  strength  and  the  direction 
of  an  electric  current  (Articles  66  and  67)  are  then  directly  ap- 
plicable. The  direction  of  the  current  at  any  point  of  a  conductor 
is  then  the  direction  of  the  current  filament  at  that  point;  the 
positive  sense  of  the  filament  is  taken  arbitrarily  as  the  direc- 
tion of  the  current.  The  strength  of  the  current  in  any  filament 
is  the  same  at  every  cross  section  of  the  filament.  Let  di  be  the 
current  in  any  filament  and  let  the  cross  section  of  this  filament 
at  any  point  P  be  dsn,  then  the  current  density  at  P  is 

di  (23) 


<T  =• 


dsn 

The  total  strength  of  the  current  across  any  surface  of  area  S  is 
then 

i=  I  (a- cos  a)  ds  (23a) 

J  s 

where  ds  is  any  element  of  this  surface,  and  a  is  the  angle  between 
the  direction  of  the  current  at  ds  and  the  normal  to  ds.  Compare 
with  equations  (16),  of  Chapter  II. 

The  results  of  all  known  experiments  show  that  the  stream 
lines  of  an  electric  current  must  be  considered  as  closed  loops 
without  ends;  in  this  respect  these  stream  lines  are  analogous 
to  lines  of  induction.  In  the  case  of  a  continuous  current,  i.e., 
a  current  which  does  not  vary  with  time,  these  stream  lines  are 
confined  entirely  to  conductors.  The  stream  lines  of  a  variable 
current,  however,  may  be  partly  or  entirely  in  a  dielectric,  even 
though  this  dielectric  be  a  perfect  insulator;  the  stream  lines  in 
a  dielectric  represent  the  displacement  current  (see  Chapter  V). 

It  should  be  noted  that  the  direction  of  the  stream  lines  in 
any  body  depends  upon  the  position  of  the  points  of  connection 
of  the  body  to  the  rest  of  the  circuit.  These  stream  lines  always 
run  through  the  body  in  the  general  direction  of  the  line  connecting 
the  two  points  of  contact  or  terminals.  The  stream  lines  will 
not  in  general  be  parallel  to  one  another,  particularly  when  the 
distance  between  the  terminals  is  large  compared  with  the  linear 
dimensions  of  the  cross  section  of  the  body  perpendicular  to 
the  direction  of  these  lines.  However,  as  has  already  been 
noted,  in  the  case  of  a  long  wire,  a  long  rod  or  strip,  or  a  long 
section  of  a  rail,  these  lines,  except  for  a  negligible  distance  in 


CONTINUOUS  ELECTRIC  CURRENTS  167 

the  immediate  vicinity  of  the  terminals,  are  practically  parallel 
to  the  axis  of  the  conductor  when  the  conductor  is  of  uniform 
cross  section  and  the  terminals  are  at  the  two  ends  of  the  con- 
ductor. Under  these  conditions  the  current  density  is  also 
uniform  over  the  cross  section  of  the  conductor  provided  the 
current  does  not  vary  with  time. 

In  the  case  of  a  variable  current  the  back  electromotive 
force  set  up  by  the  varying  magnetic  field  due  to  the  current 
is  greater  in  those  filaments  in  the  interior  of  the  conductor  than 
in  those  near  the  surface,  and  as  a  consequence  the  current  density 
is  greater  near  the  surface  of  the  conductor;  this  phenomenon  is 
known  as  the  "  skin  effect."  (See  Article  121).  In  the  case  of 
copper  or  aluminum  wires  of  the  size  ordinarily  employed  in 
practice  this  skin  effect  is  not  appreciable  except  for  rapidly  vary- 
ing currents  ;  in  the  case  of  rapidly  alternating  currents  (frequency 
greater  than  60  cycles  per  second)  this  effect  may  be  quite  appreci- 
able. In  the  case  of  steel  rails  the  skin  effect  produces  a  con- 
siderable increase  in  the  apparent  resistance  of  the  rail  even  when 
frequencies  as  low  as  15  cycles  per  second  are  employed. 

102.  Resistance  Drop  and  Electric  Intensity.  —  Electric  Equi- 
potential  Surfaces.  —  Consider  an  elementary  length  dl  of  a  current 
filament  at  any  point  P  of  a  conductor.  Let  ds  be  the  cross 
section  of  this  filament  at  the  point  P.  Then,  from  equation  (11), 
the  resistance  of  the  elementary  length  dl  is 

dl 
dR=p  — 

Hds 

where  p  is  the  specific  resistance  of  the  conducting  material. 
The  resistance  drop  in  the  length  dl  due  to  a  current  di  in  this 
filament  is  then 

dvr  =dR.di  =p  dl  — 
ds 

But  —  is  equal  to  the  current  density  cr  at  the  point  P.     Hence 
ds 

the  resistance  drop  per  unit  length  of  the  filament  at  any  point 
Pis 


The  resistance  drop  per  unit  length  of  a  current  filament  at  any 
point  is  called  the  electric  intensity  at  this  point.     This,  relation 


168  ELECTRICAL  ENGINEERING 

between  current  density  and  electric  intensity  is  of  exactly  the 
same  form  as  the  relation  between  magnetic  field  intensity  and 
flux  density  (see  equation  20  of  Chapter  II) ;  the  reciprocal  of  the 
electric  resistance,  i.e.,  the  specific  conductivity,  is  exactly  anal- 
ogous to  magnetic  permeability. 

The  c.  g.  s.  electromagnetic  unit  of  electric  intensity  is  the 
abvolt  per  centimeter ;  the  practical  unit  is  the  volt  per  centimeter 
or  the  volt  per  inch.  Electric  intensity  is  also  sometimes  expressed 
as  so  many  c.  g.  s.  electrostatic  units  per  centimeter  length. 
These  units  are  related  to  one  another  as  follows  : 

1  abvolt  per  centimeter  =  10"8  volts  per  centimeter 
=2.54  x  10"8  volts  per  inch 
1  c.  g.  s.  electrostatic 

unit  per  centimeter     =300  volts  per  centimeter 
The  resistance  drop  in  an  elementary  length  dl  of  a  current 
filament  may  then  be  written     Hedl    and    therefore    the    total 
resistance  drop  between  any  two  points  1  and  2  on  the  same 
current  filament  is 


(25) 

and  the  total  resistance  drop  between  any  two  points  1  and  2, 
whether  on  the  same  or  different  current  filaments,  is 

/2 
(  He  cos  0}  dl  (25a) 

where  He  is  the  electric  intensity  at  the  elementary  length  dl 
of  the  path  between  1  and  2  and  0  is  the  angle  between  the 
direction  of  the  electric  intensity  at  dl  and  the  direction  of  dl. 
Compare  with  equation  (25)  of  Chapter  II. 

When  there  is  no  contact  or  induced  electromotive  force  in 
the  path  from  1  to  2,  this  resistance  drop  is  equal  to  the  total 
potential  drop  along  this  path ;  when  there  is  a  contact  or  induced 
electromotive  force  e,2  in  the  path  from  1  to  2  the  total  drop  of 
potential  from  1  to  2  is 

/2 
(  He  cos  0)  dl  -  el2  (256) 

Since  the  total  electric  potential  drop  around  any  closed  path  is 
zero,  this  reduces  to 

/      (Hcos0}dl=^e  (25c) 

for  a  closed  path;  that  is,  the  total  resistance  drop  around  any 


CONTINUOUS  ELECTRIC  CURRENTS 


169 


closed  path  is  equal  to  the  total  electromotive  force  in  this  path, 
which  is  simply  another  way  of  stating  Kirchhoff's  second  law. 

Any  surface  all  points  of  which  are  at  the  same  electric  poten- 
tial is  called  an  electric  equipotential  surface.  Such  a  surface  is 
perpendicular  at  each  point  to  the  current  filament  through  that 
point  and  is  therefore  perpendicular  to  the  electric  intensity  at 
that  point.  Compare  with  magnetic  equipotential  surfaces. 

103.  Insulation  Resistance  of  a  Single  Conductor  Cable.  — 
As  an  example  of  the  use  of  the 
above  conceptions,  take  the  prob- 
lem of  calculating  the  insulation 
resistance  of  a  single  conductor 
cable  in  a  lead  sheath.  Fig.  62 
represents  the  cross  section  of  such 
a  cable.  When  all  points  of  the 
wire  are  at  the  same  potential  and 
all  points  of  the  sheath  are  at  the 
same  potential,  the  surface  of  the 
wire  and  the  inside  surface  of  the 
sheath  are  equipotential  surfaces.  Fig.  62. 

Hence  the  stream  lines  of  the  current  through  the  insulation 
(which,  it  is  to  be  remembered,  is  never  a  perfect  insulator  but 
only  an  extremely  poor  conductor)  must,  from  symmetry,  be  radial 
lines.  Let 

/  ^length  ofj  cable  in  centimeters 

rl  =  radius  of  wire  in  centimeters 

r.2=  internal  radius  of  sheath  in  centimeters 

7=total  current  in  amperes  through  the  insulation,  i.e.,  I 
is  the  leakage  current,  not  the  main  current  through  the 
wire 

V=  difference  of  potential  in  ab volts  between  the  wire  and 
the  sheath,  assumed  constant. 

p=the   specific    resistance    of    the   insulation   in  ohms  per 

centimeter-cube. 

Then  the  current  density  at  any  point  in  the  insulation  at  a  dis- 
tance x  from  the  center  of  the  wire  is 


2-rrxl 


whence  the  electric  intensity  at  this  point  is 


170 


ELECTRICAL  ENGINEERING 


=  P<T 


27TXI 

and  therefore  the  difference  of  potential  between  the  wire  and  the 
sheath  is* 


2  77  I J   TI    X 

The  insulation  resistance  of  the  given  length  of  cable  is  then 


2  -rrl 


In  r-l 


(26) 


Note  that  the  insulation  resistance  varies  inversely  as  the 
length;  this  is  evidently  true  when  it  is  remembered  that  each 
elementary  length  of  the  insulation  of  the  cable  (measured  along 
the  axis  of  the  wire)  is  in  parallel  with  all  the  other  elementary 
lengths. 

Compare  equation  (26)  with  the  formula,  equation  (20)  of 
Chapter  V,  for  the  electrostatic  capacity  of  a  sheathed  cable. 

It  can  be  shown  that  in  every  case  the  formula  for  the  insula- 
tion conductance  (i.e.,  the  reciprocal  of  the  insulation  resistance) 
between  any  two  conductors  is  identical  with  the  formula  for  the 

electrostatic  capacity  of  these  two  conductors  when  —  ^-  is  sub- 

P 

stituted  for  the  dielectric  constant  K  in  the  formula  for  capacity. 
See  Article  152  for  various  capacity  formulas. 

104.  Field  Intensity  at  any  Point  Due  to  a  Current  in  a  Wire 
of  Circular  Cross  Section.  —  Another  application  of  the  conception 
of  current  filaments  is  the  proof  of  equations  (4)  and  (4a)  of  Article 
73,  for  the  case  of  a  solid  wire  of  circular  cross  section.  A  rigid 
proof  of  these  equations  requires  the  use  of  a  geometrical  theorem 

A 


Fig.  63. 

known  as  the  Theorem  of  Inverse  Points.     This  theorem  may  be 
stated  thus:    let  P  be  any  point  either  inside  or  outside  a  circle 

*Zn  stands  for  the  natural  logarithm. 


CONTINUOUS  ELECTRIC  CURRENTS 


171 


of  radius  a  at  a  distance  r  from  the  center  0  of  the  circle,  and  let 

a2 
the  point  Q  be  on  the  line  0  P  at  a  distance  —  from  the  center  0 

on  the  same  side  of  0  as  P.  Then  the  ratio  of  the  distances  of 
any  point  A,  on  the  circumference  of  the  circle,  from  Q  and  P 

respectively,  has  the  constant  value  -•     The  two  points  P  and  Q 

are  called  inverse  points  with  respect  to  the  circle. 

To  prove  this  theorem,  draw  from  A  the  line  A  Q,  making  the 
angle  a  =  OP  A  with  the  radius  OA;  we  wish  to  show  that  the  point 

Q  where  this  line  cuts  OP  is  such  that  0  Q  =—  and  —  —  =—     Since 

r  XT 

the  two  triangles  0  QA  and  OA  P  have  one  angle  in  common  and  a 
second  angle  equal,  they  are  similar.  Hence  their  corresponding 
sides  must  be  proportional,  that  is 


or 


OP~OA~AP 


QA 

x 


whence  OQ=—  and  --  =—  .  a  constant.     When  the  point   P  is 
r  x       r 

outside  the  circle,  Fig.  63,  its  inverse  point  Q  is  inside  the  circle  and 


Fig.  64. 


when  the  point  P  is  inside  the  circle,  Fig.  64,  its  inverse  point  Q 
is  outside  the  circle.     This   follows   immediately  from  the  fact 

that  OQ=aY-)and  therefore  when  r  is  greater  than  a,  OQ  is  less 

than  a,  and  when  r  is  less  than  a,  0  Q  is  greater  than  a. 

Now  consider  a  straight  hollow  wire  or  tube  of  infinite  length 
and  let  the  walls  of  this  tube  be  of  infinitesimal  thickness  t.  Let 
a  be  the  radius  of  this  tube  and  I  the  total  current  flowing  in  its 
walls.  These  walls  may  be  considered  as  made  up  of  filaments  of 
infinitesimal  cross  section  tds  where  ds  is  an  elementary  length 


172 


ELECTRICAL  ENGINEERING 


in  the  circumference  of  the  circle  representing  the  cross  section  of 
the  tube.  When  the  current  is  uniformly  distributed  in  the 
walls  of  the  tube,  the  current  in  each  filament  is 

ids  I  ds 

dl==2^t     ~~2^ 

Let  P  be  any  point  at  a  perpendicular  distance  r  from  the  axis 
of  the  tube,  and  let  x  be  the  perpendicular  distance  of  P  from  a 
filament  A,  Then  from  equation  (34a)  this  filament  produces  a 
field  intensity 


X  TTCL      X 

in  the  direction  perpendicular  to  A  P.  (The  direction  of  the  cur- 
rent is  assumed  up  toward  the  reader.)  This  intensity  may  be 
resolved  into  two  components,  one  parallel  to  OP  and  the  other 
perpendicular  to  OP.  A  second  filament  ds'  of  equal  cross  sec- 


Fig.  65. 

tion  symmetrically  located  at  A'  on  the  other  side  of  the  line  OP 
will  produce  an  equal  field  intensity  dH'  at  P  in  the  Direction 
perpendicular  to  A' P.  Its  component  parallel  to  OP  will  be 
equal  and  opposite  to  the  component  of  dH  in  this  direction. 
Similarly  for  any  other  pair  of  symmetrically  located  filaments. 
Hence  the  resultant  field  intensity  at  P  due  to  the  entire  tube  is 
equal  to  the  sum  of  the  components  perpendicular  to  OP  of  the 
field  intensities  due  to  all  the  filaments,  and  will  be  in  the  direction 
perpendicular  to  OP. 

The  component  perpendicular  to  OP  of  the  field  intensity  at 
P  due  to  the  filament  ds  is 


dHn=— 
n     TT  a 


cos  a 
x 


Let  Q  be  the  inverse  point  of  P  with  respect  to  the  circle  represent- 
ing the  cross  section  of  the  tube,  and  let  ft  represent  the  angle 
P  Q  A  and  d  ft  the  angle  at  Q  subtended  by  the  arc  ds.  Draw  a 
line  AB  through  A  perpendicular  to  QA;  this  line  will  make  the 
angle  a  with  ds,  since  the  angle  between  OA  and  QA  is  a,  and  OA 


CONTINUOUS  ELECTRIC  CURRENTS  173 

is  perpendicular  to  ds  and  QA  to  AB.     Hence  the  projection  of 
ds  on  this  line  A  B  is  (ds  cos  a).     Hence,  since  ds  is  infinitesimal, 


d  ^      But  gince   Q  and  p  are  inyerse       intg          =  __ 

QA  x         r 

ax     r™       r         jo 
whence  QA  =  —  .    Therefore  dp= 
r 

ds  cos  a_a     _ 
x       ~r     " 
Hence 

dHn=—  -    dft=-i- 
TTa  r  TT  r 

and  therefore  the  total  field  intensity  at  P  when  P  is  outside  the 
tube  is 


J0=27T  />=2»   2/ 

dtfn=j:     d£=7 
0=o         ^J  *£      r 


Therefore  the  field  intensity  at  any  point  outside  a  tube  of  circular 
cross  section  carrying  a  current  I  is  the  same  as  would  be  pro- 
duced by  a  current  of  the  same  strength  concentrated  in  a  line 
coinciding  with  the  center  of  the  tube.  Since  a  solid  wire  of  cir- 
cular cross  section  may  be  considered  as  made  up  of  a  series  of 
concentric  tubes,  this  formula  is  likewise  true  for  any  point  P 
outside  a  solid  circular  wire. 

When  the  point  P  is  inside  the  tube  the  inverse  point  Q  is  out- 
side. Hence  in  this  case  the  two  limiting  values  of  the  angle  ft 
are  identical  and  equal  to  TT;  therefore  the  total  field  intensity  at 
Pis 


/>« 

TrJ    ^= 

J     /3  =  7T 


Trr 
i 

That  is,  the  field  intensity  at  any  point  inside  a  circular  tube  in 
which  the  current  is  uniformly  distributed  is  zero. 

At  any  point  P  inside  a  solid  circular  wire  in  which  the  current 
is  uniformly  distributed  the  field  intensity  is  therefore  due  only 
to  the  current  inside  the  cylinder  the  cross  section  of  which  is  the 
circle  through  P  concentric  with  the  center  of  the  wire.  Calling  r  the 
distance  of  the  point  from  the  center  of  the  wire,  a  the  radius  of 
the  wire  and  /  the  total  current,  the  current  in  this  cylinder  is  then 

?rr2        r* 
— 2  7  ==—  I,  since  ?rr2  is  the  area  of  the  cross  section  of  the  cylinder 

and  Tra2  is  the  area  of  the  cross  section  of  the  whole  wire.     Hence 
the  field  intensity  at  the  point  P  inside  the  wire  is 


174  ELECTRICAL  ENGINEERING 

That  is,  at  any  point  inside  the  solid  circular  wire  the  field  inten- 
sity varies  directly  as  the  distance  of  the  point  from  the  center  of 
the  wire,  while  outside  the  wire  the  field  intensity  varies  inversely 
as  the  distance  of  the  point  from  the  center  of  the  wire. 

SUMMARY  OF  IMPORTANT  DEFINITIONS  AND 
PRINCIPLES 

1.  As  the  measure  of  the  strength  of  an  electric  current  in  a 
wire  is  taken  the  ratio  of  the  force  per  unit  length  of  the  wire, 
which  would  be  produced  on  the  wire  by  a  magnetic  field,  to 
the  component  of  the  flux  density  of  this  field  perpendicular  to 
the  wire.     The  c.  g.  s.  electromagnetic  unit  of  current  strength 
is  the  abampere.     The  practical  unit  is  the  ampere 

1  abampere  =10  amperes. 

2.  Left-hand  Rule.  The  direction  of  the  current  (I)  in  a  wire 
is  the  direction  in  which  the  middle  finger  of  the  left  hand  points 
when  the  thumb,  forefinger  and  middle  finger  of  this  hand  are 
held  mutually  perpendicular  and  the  thumb  is  pointed  in  the 
direction  in  which  the  wire  tends  to  move  and  the  forefinger  is 
pointed  in  the  direction  of  the  component  of  the  flux  density 
perpendicular  to  the  wire. 

3.  A  continuous  electric  current  is  a  current  the  strength  of 
which  does  not  vary  with  time. 

4.  A  conductor  is  a  substance  in  which,  when  connected  to 
the  poles  of  a  battery,  a  continuous  electric  current  is  established. 
An  insulator  or  dielectric  is  a  substance  in  which,  when  connected 
to  the  poles  of  a  battery,  no  continuous  current,  or  only  a  very 
small  continuous  current,  is  established. 

5.  Two  or  more  conductors  connected  end  to  end  in  such  a 
manner  that  the  same  current  flows  through  each  are  said  to  be 
connected  in  series.     Two  or  more  conductors  joining  any  two 
points  of  an  electric  circuit  in  such  a  manner  that  the  total  current 
entering  these  conductors  at  one  point  leaves  these  conductors 
at  the  other  point  are  said  to  be  connected  in  parallel. 

6.  The  mechanical  force  acting  on  a  wire  I  centimeters  long 
carrying  a  current  of  7  abamperes  due  to  the  magnetic  field  in 
which  the  wire  is  placed  is 

—7-1 
F  =        I  (B  sin  9}  dl  dynes 

J     0 

where  B  is  the  flux  density  in  gausses  at  any  elementary  length 


CONTINUOUS  ELECTRIC  CURRENTS  175 

dl  of  the  wire  and  0  is  the  angle  between  the  direction  of  B  and 
the  direction  of  dl.  For  a  straight  wire  in  a  uniform  field  this 
reduces  to 

F=IBlsin  9  dynes. 

7.  The  magnetic  field  intensity  at  any  point  P  due  directly  to 

a  current  of  7  abamperes  in  a  wire  I  centimeters  long  is 

u 

(7  sin  6}    ,,  .,, 

-  dl  gilberts  per  cm. 


- 

J  i 


where  6  is  the  angle  between  dl  and  the  line  drawn  from  P  to  dl 
and  r  is  the  distance  from  P  to  dl.  The  field  intensity  due  directly 
to  the  current  is  independent  of  the  magnetic  nature  of  the  bodies 
in  the  field;  if  any  magnetic  poles  are  induced  by  this  field  the 
field  intensity  due  to  these  poles  must  be  added  (vectorially)  to 
the  above. 

The  magnetic  field  intensity  at  a  point  due  to  a  current  of  1 
abamperes  in  a  very  long  straight  wire  of  circular  cross  section  at 
a  point  r  centimeters  from  the  wire  is 

27 

77= —  gilberts  per  cm. 

r 

when  the  point  is  outside  the  wire.  When  the  point  is  inside  a  wire 
having  a  radius  of  a  centimeters  and  the  wire  is  solid,  the  field 
intensity  is 

77= —  gilberts  per  cm. 

a2 

provided  the  current  density  in  the  wire  is  uniform. 

The  field  intensity  at  the  center  of  a  circular  coil  with  a  con- 
centrated winding  of  N  turns  carrying  a  current  of  7  abamperes  is 

u       2  77   N  I  ... 

H  =—  gilberts  per  cm. 

r 

where  r  is  the  mean  radius  of  the  coil  in  centimeters. 

8.  The  lines  of  magnetic  induction  due  to  an  electric  current 
are  always  closed  loops  linking  the  current.     The  positive  sense 
of  these  lines  is  the  same  as  the  direction  in  which  a  right-handed 
screw  placed  along  the  wire  must  be  turned  to  advance  the  screw 
in  the  direction  of  the  current. 

9.  Any  substance  the  constituents  of  which  are  separated  when 
an  electric  current  is  established  in  it  is  called  an  electrolyte  and 
the  process  of  separation  is  called  electrolysis.     The  terminal  at 
which  the  current  enters  an  electrolyte  is  called  the  anode  and  the 


176  ELECTRICAL  ENGINEERING 

terminal  from  which  the  current  leaves  the  electrolyte  is  called 
the  cathode;  the  name  electrode  is  used  for  either  terminal. 

10.  The  relation  between  the  mass  m  of  a  substance  deposited 
in  time  t  by  a  continuous  current  I  and  this  current  is 

2-tf 
t 

where  &  is  a  constant,  called  the  electrochemical  equivalent  of  the 
substance,  which  constant  depends  on  the  nature  of  the  substance 
and  the  units  in  which  the  various  quantities  are  measured.  For 
a  variable  current  this  formula  becomes 

dm 

—  =ki 
dt 

The  electrochemical  equivalent  of  a  substance  is  proportional  to 
its  chemical  equivalent. 

11.  The  quantity  of  electricity  Q  which  flows  through  any  sec- 
tion of  a  conductor  is  defined  as  the  product  of  the  strength  /  of 
the  current  in  this  section  by  the  time  t  during  which  the  current 
flows,  i.e., 

Q=It 

This  applies  only  to  a  continuous  current ;  in  the  case  of  a  variable 
current  the  quantity  is 

Q=f  idt 

J    o 

When  the  current  is  expressed  in  abamperes  and  the  time  in  sec- 
onds the  unit  of  quantity  is  the  abcoulomb;  when  these  quantities 
•are  expressed  in  amperes  and  seconds  respectively,  the  unit  is  the 
coulomb. 

1  abcoulomb  =10  coulombs 

12.  The  electric  resistance  of  a  conductor  is  defined  as  the  ratio 
of  the  rate  at  which  a  continuous  current  produces  heat  energy  in 
the  conductor  to  the  square  of  the  strength  of  this  current,  pro- 
vided the  conductor  is  of  uniform  structure  and  is  kept  at  uniform 
temperature  throughout.     The  power  dissipated  in  a  conductor  by 
a  continuous  current  of  strength  7,  due  to  the  resistance  R  of  the 
conductor  is  _ 

Ph  =  RP 

This  formula  gives  the  power  in  ergs  per  second  when  the  current 
7  is  expressed  in  abamperes  and  the  resistance  R  is  expressed  in 
c.  g.  s.  electromagnetic  units  or  abohms ;  when  the  current  is  ex- 


CONTINUOUS  ELECTRIC  CURRENTS  177 

pressed  in  amperes  and  the  resistance  in  ohms  (the  practical  unit) 
this  formula  gives  the  power  in  watts. 

1  ohm=109  abohms 

13.  The  resistance  of  a  conductor  of  length  /and  cross  section 
A  is 

,.,< 

where  the  factor  p,  called  the  specific  resistance  of  the  conductor, 
is  a  constant  for  a  given  material  at  a  given  temperature.  This 
formula  applies  only  when  the  stream  lines  of  the  current  are 
parallel  and  perpendicular  to  the  section  A  and  the  current  density 
is  uniform  over  this  section.  When  p  is  expressed  in  ohms  per 
centimeter-cube  the  dimensions  of  the  conductor  must  be  expressed 
in  centimeters  ;  when  p  is  expressed  in  ohms  per  inch-cube  the 
dimensions  must  be  expressed  in  inches  ;  when  p  is  expressed  in 
ohms  per  mil-foot  the  length  must  be  expressed  in  feet  and  the 
cross  section  in  circular  mils. 

14.  The  reciprocal  of  electric  resistance  is  called  the  electric 
conductance. 

15.  The  variation  of  the  resistance  of  a  conductor  with  tem- 
perature is  expressed  by  the  formula 


where  R  is  the  resistance  at  zero  degrees  centigrade,  R  the  resist- 
ance at  any  other  temperature  t  degrees  centigrade,  and  ft  is  the 
temperature  coefficient  of  resistance.  The  temperature  coefficient 
of  copper  is  0.0042  ;  the  temperature  of  other  pure,  non-magnetic 
conductors  has  approximately  the  same  value.  When  this  value 
of  the  temperature  coefficient  is  employed  the  relation  between  the 
resistances  Rt  and  Rt'  at  any  two  temperatures  t  and  if  degrees 
centigrade  is  given  by  the  formula 


238  +  t 

16.  The  electric  energy  gained  by  the  current  in  any  part  of  a 
circuit  is  defined  as  the  amount  of  energy  lost  by  this  part  of  the 
circuit  or  by  any  other  bodies  as  the  result  of  the  existence  of  the 
current  in  this  part  of  the  circuit  ;  the  electric  energy  lost  by  the 
current  in  any  part  of  a  circuit  is  defined  as  the  amount  of  energy 
gained  by  this  portion  of  the  circuit  or  any  other  bodies  as  the 
result  of  the  existence  of  the  current  in  this  part  of  the  circuit. 


178  ELECTRICAL  ENGINEERING 

17.  The  drop  of  electric  potential  from  any  point  1  to  any  point 
2  along  a  wire  carrying  an  electric  current  is  defined  as  the  ratio 
of  the  power  developed  by  the  current  between  these  two  points 
to  the  value  of  the  current  from  1  to  2.      Hence  the  power  devel- 
oped by  a  current  I  (i.e.,  the  rate  at  which  electric  energy  is  lost 
by  the  current)  in  any  portion  of  a  circuit  in  which  the  drop  of 
potential  in  the  direction  of  the  current  is  V  is 

P  =  VI 

This  formula  gives  the  power  in  ergs  per  second  when  the  current 
/  is  expressed  in  abamperes  and  the  potential  drop  V  is  expressed 
in  c.  g.  s.  electromagnetic  units  or  abvolts  ;  when  the  current  is 
expressed  in  amperes  and  the  potential  drop  in  volts  (the  practical 
unit)  this  formula  gives  the  power  in  watts 
1  volt=108  abvolts. 

18.  The  drop  of  potential  in  a  wire  due  solely  to  the  resistance 
of  the  wire,  or  the  resistance  drop,  is 

Vr=RI  volts 

where  all  quantities  are  in  practical  units. 

19.  An  electromotive  force  is  that  which  produces  or  opposes 
the  flow  of  electricity,  other  than  the  opposition  due  to  the  resist- 
ance of  the  conductor  in  which  it  flows.      As  the  measure  of  the 
electromotive  force  in  any  portion  of  the  circuit  is  taken  the  rise 
of  potential  which  it  would  produce  in  the  direction  of  the  current 
in  this  portion  of  the  circuit  were  there  no  resistance  drop  in  this 
portion  of  the  circuit.     When  the  electromotive  force  produces  an 
actual  rise  of  potential  in  the  opposite  direction  to  that  of  the 
current  it  is  called  a  back  electromotive  force. 

20.  The  net  rise  of  potential  E'l2  from  terminal  1  to  terminal  2 
of  any  circuit  is  called  the  terminal  electromotive  force  of  the  cir- 
cuit when  the  current  in  this  portion  of  the  circuit  is  from  1  to  2  ; 
when  the  current  in  this  portion  of  the  circuit  is  in  the  direction 
from  2  to  1  the  net  rise  of  potential  E'l2  from  1  to  2  is  called  the 
impressed  electromotive  force,  due  to  the  rest  of  the  circuit.     The 
relation  between  the  electromotive  force   E12  generated  in  this 
portion  of  the  circuit,  the  current  712  in  this  portion  of  the  circuit, 
the  terminal  or  impressed  electromotive  force  E'ia  and  the  resist- 
ance R  of  this  portion  of  the  circuit  is 


R 


CONTINUOUS  ELECTRIC  CURRENTS  179 

where  all  quantities  are  expressed  in  practical  units.     This  relation 
is  known  as  the  Generalized  Ohm's  Law. 

21.  In  general,  an  electromotive  force  exists  at  the  surface  of 
contact  of  any  two  dissimilar  substances,  and  the  value  of  this 
electromotive  force  of  contact  depends  only  upon  the  nature  of  the 
substances  in  contact  and  upon  the  temperature  of  the  junction. 

22.  Contact  electromotive  forces  are  appreciable  only  when  the 
substances  in  contact  act  chemically  upon  each  other  or  when 
there  are  two  junctions  in  the  circuit  kept  at  different  tempera- 
tures.    The  electromotive  force  of  a  chemical  battery  is  due  to  the 
contact  electromotive  forces  between  the  substances  of  which  the 
battery  is  made.     The  electromotive  force  of  a  thermocouple  is 
due  to  the  difference  in  the  electromotive  forces  at  the  hot  and 
cold  junctions. 

23.  KirchhofFs  Laws.  —  These  so-called  "  laws  "  are  the  two 
experimental  facts  : 

a.  The  algebraic  sum  of  the  currents  coming  up  to  any  junction 
in  a  network  of  conductors  is  equal  to  zero,  i.e., 

27-0 

b.  The  algebraic  sum  of  the  resistance  drops  around  any  closed 
loop  in  a  network  of  conductors  is  equal  to  the  algebraic  sum  of 
the  electromotive  forces  in  this  loop,  i.e., 

2  E=tRI 

24.  The  total  resistance  of  two  or  more  conductors  in  series  is 
equal  to  the  sum  of  the  resistances  of  these  conductors.     The 
resultant  electromotive  force  of  two  or  more  electromotive  forces 
in  series  is  the  algebraic  sum  of  these  electromotive  forces. 

25.  The  total  conductance  of  two  or  more  conductors  in  par- 
allel is  equal  to  the  sum  of  the  conductances  of  these  conductors, 
provided  there  are  no  electromotive  forces  in  these  conductors. 
The  resultant  resistance  of  two  conductors  in  parallel,  when  there 
are  no  electromotive  forces  in  these  conductors,  is 


The  resultant  electromotive  force  of  two  or  more  equal  electromo- 
tive forces  in  parallel  is  the  same  as  each  electromotive  force. 

26.  A  conductor  of  any  shape  in  which  an  electric  current  is 
established  may  be  divided  into  filaments  separated  from  each 
other  by  insulating  walls  of  infinitesimal  thickness  without  alter- 
ing any  of  the  effects  produced  in  or  around  the  conductor,  pro- 


180 


ELECTRICAL  ENGINEERING 


vided  these  filaments  are  drawn  in  the  proper  directions.     Such 
filaments  are  called  the  stream  lines  of  the  electric  current. 

27.  The  electric  intensity  at  any  point  in  a  conductor  coincides 
in  direction  with  the  stream  line  through  that  point  and  is  equal 
to  the  product  of  the  specific  resistance  p  of  the  conductor  and  the 
current  densitv  cr  at  that  point,  i.e., 

B.-P.* 

The  resistance  drop  between  any  two  points  1  and  2  in  a  conductor 
of  any  shape  is 


••=/: 


(  He  cos  6}  dl 


where  He  is  the  electric  intensity  at  the  element  dl  of  the  path 
between  1  and  2  and  9  is  the  angle  between  the  electric  intensity 
at  dl  and  the  direction  of  dl. 

28.  Electric  equipotential  surfaces  and  stream  lines  of  electric 
current  are  mutually  perpendicular. 

29.  The  formula  for  the  insulation  conductance  between  any 
two  conductors  is  identical  with  the  formula  for  the  electrostatic 
capacity  between  these  two  conductors  when  4  TT  divided  by  the 
specific  resistance  p  is  substituted  for  the  dielectric  constant   K 
in  the  formula  for  capacity.  *  £ 

PROBLEMS 

1.  A  straight  wire  carrying  a  current  of  20  amperes  is  placed 
in  the  gap  between  the  poles  of  two  magnets,  the  pole-faces  of 
B  c      which  are  each   1   inch  square. 

If  the  flux  in  the  air  gap  may  be 
represented  by  straight  lines  and 
the  flux  density  is  1000  gausses, 
calculate  the  force  in  dynes  act- 
ing upon  the  wire,  when  the  wire 
is  perpendicular  to  the  flux  lines. 
Ans.:  5080  dynes. 
2.  Given  the  electrical  circuit 
shown  in  the   figure.      The  wire 
at  A  is  insulated  so  that  all  the 
current  flowing  at  F  must  pass 
around   the    circular   loop    in    a 
counter-clockwise    direction    be- 
fore continuing  to  B.      The  current  may  be  assumed  to  flow  in 


CONTINUOUS  ELECTRIC  CURRENTS  181 

a  geometrical  line  in  all  parts  of  the  circuit  and  in  the  battery. 
Dimensions:  Radius  OA  =2  inches,  AB  =6  inches,  BC  =  4 
inches,  CE  =  8  inches,  EF  =  4:  inches,  and  FA  =2  inches.  If 
the  current  from  B  to  C  is  50  amperes,  find  the  field  magnetic 
intensity  at  0  due  to  the  entire  circuit. 
Ans.:  7.11  gilberts  per  cm. 

3.  A  current  of  100  amperes  is  established  in  a  long  iron  rod 
the  diameter  of  which  is  1  inch.     The  rod  has  a  permeability  of 
300  c.  g.  s.  units.     What  is  the  magnetic  flux  density  0.25  inch 
from  the  center  of  the  rod  due  to  this  current?     What  is  the  total 
number  of  lines  of  induction  per  foot  length  inside  the  rod? 

Ans.:  2360  gausses.     91,400  maxwells. 

4.  The  current  established  in  a  solution  of  copper  sulphate 
(CuS04)  is  100  amperes.     Determine  the  weight  of  copper  de- 
posited on  a  platinum  cathode  in  one  hour.     What  quantity  of 
electricity  is  transferred  through  the  solution? 

Ans.:  118.7  grams.     100  ampere-hours  or  360,000  coulombs. 

5.  A  copper  wire  two  miles  in  length  has  a  cross  section  of 
0.07  square  inch  and  has  a  resistance  of  1.13  ohms.     What  is  the 
length  in  feet  of  a  wire  of  the  same  material  20,000  circular  mils 
in  cross  section  and  having  a  resistance  of  1.5  ohms? 

Ans.:  3150  feet. 

6.  The  resistance  per  mil-foot  of  copper  is  9.59  ohms  at  0°  cent. 
What  will  be  the  resistance  at  this  temperature  of  40  grams  of 
copper  wire  which  has  a  cross  section  of  2500  circular  mils  and 
a  specific  gravity  of  8.89? 

Ans.:  0.0447  ohms. 

7.  The  resistance  of  the  armature  winding  of  a  given  electric 
motor  at  24°  cent,  is  found  to  be  1.702  ohms.     The  armature 
resistance  is  again  measured  after  the  motor  has  been  in  service 
for  several  hours  and  found  to  be  1.980  ohms.     What  is  the  aver- 
age increase  in  the  temperature  of  the  armature  winding? 

Ans.:  43°  cent. 

8.  A  100-volt  generator  and  a  50-volt  battery  are  connected 
in  series.     The  internal  resistance  of  the  generator  is  1  ohm  and 
the  internal  resistance  of  the  battery  7  ohms ;   the  resistance  of 
each  of  the  two  wires  connecting  the  battery  and  the  generator  is 
1   ohm.     What  is  the  current  in  this  circuit  and  the  terminal 
voltage  of  the  battery  ( 1)  when  the  electromotive  forces  of  the  bat- 
tery and  the  generator  oppose  each  other,  and  (2)  when  these  elec- 


182  ELECTRICAL  ENGINEERING 

tromotive  forces  are  in  the  same  direction  around  the  circuit? 
The  electromotive  forces  are  to  be  assumed  constant;  actually, 
the  electromotive  force  of  the  battery  will  depend  upon  the  amount 
and  direction  of  the  current,  due  to  the  polarisation  which  takes 
place.  (3)  How  much  of  the  power  developed  by  the  engine 
driving  the  generator  is  converted  into  electric  power  in  each  case? 
(4)  At  what  rate  is  energy  transformed  into  chemical  energy  in 
the  first  case?  (5)  At  what  rate  is  chemical  energy  transformed 
into  electric  energy  in  the  second  case?  (6)  At  what  rate  is  energy 
transformed  into  heat  energy  in  each  case? 

Ans.:  (1)  5  amperes  and  85  volts ;  (2)  15  amperes  and  55  volts  ; 
(3)  500  watts  and  1500  watts;  (4)  250  watts;  (5)  750  watts;  (6) 
250  watts  and  2250  watts. 

9.  A  house  service  consists  of  5  (32  C.P.)  lamps  of  110  ohms 
resistance  each,  8  (16  C.P.)  lamps  of  220  ohms  resistance  each  and 
an  electric  heater  of  10  ohms  resistance,  all  connected  in  parallel. 
The  voltage  between  the  service  wires  at  the  entrance  to  the  house 
is  115,  and  each  of  the  two  wires  leading  from  the  entrance  to  the 
load  has  a  resistance  of  0.1  ohm.     What  is  the  energy  in  kilowatt- 
hours  delivered  to  the  house  during  a  period  of  4  hours? 

Ans.:  9.28  kilowatt-hours. 

10.  Three  batteries  with  electromotive  forces  of  15,  20  and  25 
volts  respectively  and  with  internal  resistances  of  4,  3  and  2  ohms 
respectively,  have  their  positive  terminals  connected  together  and 
their  negative  terminals  connected  together.     (1)  What  is  the  ter- 
minal electromotive  force  of  each  battery,  and  (2)  what  is  the 
current  in  each? 

Ans.:  (1)  21.15  volts;  (2)  1.538  amperes,  0.385  amperes,  1.923 
amperes. 

11.  Fig.  B  represents  a  so-called  "  three  wire  system  "  which 

is  largely  used  for  distributing 
electric  energy  for  light  and 
power.  A  and  B  are  two  gene- 
rators (the  field  windings  are 
omitted  for  simplicity)  with 
their  electromotive  forces  act- 
ing in  the  same  direction.  C 
and  D  are  the  loads,  which 
*>  may  be  either  lamps  or 

rig.  B.  motors,     a  and  b  are  the  out- 


CONTINUOUS  ELECTRIC  CURRENTS  183 

side  "  mains  "  and  c  is  the  "  neutral "  wire.  (Motors  are  also 
frequently  connected  across  the  outside  mains  a  and  b.)  Let  the 
generated  electromotive  forces  of  the  two  generators  A  and  B 
each  be  110  volts,  their  internal  resistances  be  1.5  and  2  ohms 
respectively,  the  effective  resistances  of  the  loads  C  and  D  be  8 
ohms  and  10  ohms  respectively,  and  the  resistance  of  each  of  the 
three  wires  a,  b  and  c  be  0.1  ohm.  (1)  What  are  the  currents 
taken  by  the  loads  C  and  Df  (2)  What  is  the  current  in  the 
neutral  wire  c?  (3)  What  are  the  terminal  voltages  of  the 
generators  A  and  B,  and  (4)  the  impressed  voltages  at  the  load? 
Ans.:  (1)  11.43  amperes  and  9.11  amperes;  (2)  2.32  amperes; 
(3)  92.8  volts  and  91.8  volts;  (4)  91.4  volts  and  90.8  volts. 

12.  Energy  is  delivered  from  a  230- volt  generator  to  a  factory 
over  a  transmission  line  which  has  a  total  resistance  of  0.2  ohms. 
What  is  the  current,  the  potential  difference  at  the  factory  and 
the  efficiency  of  transmission  when  the  power  taken  by  the  factory 
is  50  kilowatts? 

Ans.:  There  are  two  possible  currents,  depending  upon  the 
resistance  of  the  load.  These  currents  are  291.2  amperes  and 
858.8  amperes;  the  corresponding  potential  differences  are  171.8 
volts  and  58.2  volts ;  the  corresponding  efficiencies  are  74.7%  and 
25.3%.  In  practice  the  resistance  of  the  load  is  always  such  that 
the  smaller  current  and  therefore  the  higher  efficiency  is  obtained. 

13.  Fifty  kilowatts  of  power  are  to  be  delivered  to  a  factory 
5000  feet  from  a  power  house.     The  voltage  at  the  power  house  is 
600.     What  must  be  the  cross  section  of  the  wire  in  circular  mils 
used  for  the  transmission  line  in  order  that  the  efficiency  of  trans- 
mission be  90%?     Assume  the  wire  to  have  a  conductivity  of 
98%  and  the  temperature  to  be  20°  cent.     What  would  be  the 
cost  of  the  copper  for  this  line  if  the  price  of  copper  is  20  cents 
per  pound? 

Ans.:  (1)  163,800  circular  mils.  This  corresponds  approxi- 
mately to  a  No.  000  B.  &  S.  gauge  wire.  (See  Appendix  B.)  (2) 
$1016  if  No.  000  wire  is  used. 

14.  The  inside  diameter  of  a  lead-sheathed,  rubber- insulated 
cable  which  has  a  cross  section  of  250,000  circular  mils  is  1  inch. 
If  the  rubber  has  a  specific  resistance  of  150X107  megohms  per 
centimeter-cube  at  20°  cent.,  what  is  the  insulation  resistance  per 
mile  of  this  cable  at  this  temperature?     (Cables  are  usually  made 
of  stranded  wires,  and  the  cross  section  of  the  strand  is  therefore 


184  ELECTRICAL   ENGINEERING 

not  a  perfect  circle;  in  this  problem,  however,  the  wire  may  be 
assumed  solid  and  of  circular  cross  section.) 

Ans.:  1027  megohms  per  mile. 

15.  Prove  that  the  resistance  of  a  uniformly  tapered  wire  (i.e., 
a  wire  such  that  its  diameter  changes  by  a  constant  amount  per 
unit  distance  measured  along  its  axis)  is 

R  =  n ohms 

IT  fir, 

where  p  is  the  specific  resistance  of  the  wire  per  centimeter-cube 
I  is  its  length  in  centimeters,  and  rl  and  rz  are  the  radii  in  centi- 
meters of  its  cross  section  at  its  two  ends.  The  current  density 
at  any  cross  section  of  the  wire  is  to  be  assumed  constant  over 
that  cross  section.  Compare  with  the  resistance  of  a  wire  of  uni- 
form cross  section. 


IV 
ELECTROMAGNETISM 

105.  Electromotive  Force  Due  to  Change  in  the  Number  of 
Lines  of  Induction  Linking  a  Circuit. —  Linkages  between  Flux 
and  Current.  —  In  the  last  chapter  were  described  two  types  of 
devices  for  producing  an  electric  current,  the  chemical  battery  and 
the  thermo-electric  couple.  In  engineering  work,  however,  neither 
of  these  devices  is  used  as  a  "  generator  "  of  electric  energy,  except 
in  small  amounts  for  testing  purposes  and  where  only  small  cur- 
rents are  required,  on  account  of  the  high  cost  of  generating  elec- 
trical energy  by  such  devices.  Practically  all  electric  generators 
and  all  electric  motors  are  based  upon  an  entirely  different  prin- 
ciple, namely,  that  whenever  the  number  of  lines  of  magnetic  induc- 
tion linking  a  circuit  is  changed  an  electromotive  force  is  produced  in 
this  circuit.  This  important  fact,  which  is  the  basis  of  electrical 
engineering,  was  discovered  by  Faraday  in  the  early  part  of  the 
last  century.  Electrical  engineering  as  an  art,  however,  cannot 
be  said  to  date  back  earlier  than  about  1880,  when  the  develop- 
ment of  the  incandescent  lamp  by  Edison  first  created  a  demand 
for  large  amounts  of  electric  energy. 

The  electromotive  forces  produced  in  an  electric  circuit  by 
changing  the  number  of  lines  of  induction  linking  the  circuit  are 
called  "  induced  "  electromotive  forces,  and  the  currents  due  to 
these  electromotive  forces  are  called  "  induced  "  currents. 

As  has  already  been  noted,  every  electric  circuit  forms  one  or 
more  closed  loops  and  every  line  of  induction  forms  a  closed  loop. 
A  line  of  induction  and  an  electric  circuit  may  therefore  link  each 
other  in  the  same  way  that  two  links  of  a  chain  link  each  other. 
When  the  electric  circuit  is  in  the  form  of  a  coil  of  a  number  of 
turns  the  same  line  of  induction  may  thread  several  turns  of  the 
coil;  in  this  case  the  line  of  induction  is  said  to  form  with  the 
circuit  a  number  of  linkages  equal  to  the  number  of  turns  which 
it  threads.  The  total  number  of  linkages  corresponding  to  any 
number  of  lines  of  induction  is  the  sum  of  the  linkages  of  all  the 
lines.  In  the  special  case  of  a  coil  of  N  turns  and  <f>  lines  of 

185 


186  ELECTRICAL  ENGINEERING 

induction  each  of  which  threads  all  these  turns,  the  total  number 
of  linkages  is  X  =  N  <f> ;  if  only  part  of  the  lines  thread  all  the 
turns  the  number  of  linkages  will  be  less  than  N  <£.  The  symbol 
X  will  be  used  throughout  for  number  of  linkages ;  linkages  are 
expressed  in  the  same  unit  as  flux  of  induction,  i.e.,  in  maxwells. 
With  this  understanding  the  experimentally  determined  rela- 
tion between  the  value  of  the  induced  electromotive  force  and  the 
change  in  the  number  of  lines  of  induction  linking  a  circuit  may 
be  stated  as  follows :  The  value  of  the  electromotive  force  induced  in 
an  electric  circuit  is  equal  to  the  time  rate  of  change  of  the  number 
of  linkages  between  the  circuit  and  the  lines  of  induction  threading 
it.  Or,  using  the  word  "  flux  "  as  synonymous  with  "  lines  of 
magnetic  induction,"  this  law  may  be  stated  more  briefly  as,  the 
electromotive  force  induced  in  a  circuit  is  equal  to  the  time  rate  of 
change  of  the  linkages  between  the  flux  and  the  circuit. 

The  direction  of  this  induced  electromotive  force  is  found 
to  be  such  that  it  would  of  itself  set  up  a  current  in  the  circuit 
in  such  a  direction  as  to  oppose  the  change  in  the  flux  link- 
ing the  circuit.  We  have  already  seen  (Article  72)  that  the 
direction  of  the  lines  of  magnetic  induction  set  up  by  a  cur- 
rent in  any  circuit  is  always  such  that  these  lines  link  the  cir- 
cuit in  the  direction  in  which  a  right-handed  screw  placed 
along  the  wire  forming  this  circuit  would  have  to  be  turned  to 
advance  it  in  the  direction  of  the  current.  Hence  the  direction 
of  the  electromotive  force  induced  in  a  circuit  when  the  flux  linking 
this  circuit  is  changed  is  opposite  to  the  direction  in  which  a  right- 
handed  screw  placed  along  the  conductor  forming  the  circuit 
would  advance  if  turned  in  the  direction  in  which  the  lines  of 
induction  are  increased.  Hence,  calling  d\  the  increase  in  time 
dt  of  the  number  of  linkages  between  the  circuit  and  the  flux 
threading  it,  the  induced  electromotive  force  in  the  right-handed 
screw  direction  with  respect  to  the  direction  of  the  lines  of  induc- 
tion linking  this  circuit  is 

_d\  (1) 

~dt 

Where  X  is  expressed  in  maxwells  or  c.  g.  s.  lines  of  induction  and 
t  in  seconds,  this  equation  gives  the  value  of  the  induced  electro- 
motive force  in  abvolts.  The  value  of  the  induced  electromotive 
force  in  volts,  when  X  is  expressed  in  maxwells  and  t  in  seconds,  is 
since  1  volt  =  108  abvolts 


ELECTROMAGNETISM  187 


dt 

When  the  circuit  in  question  consists  of  a  single  loop  the  link- 
ages X  are  equal  to  the  flux  <£,  and  the  induced  electromotive 
force  is  then 

e=-dA  abvolts  =  -  ID'8  ^  volts  (16) 

dt  dt 

When  the  circuit  in  question  is  formed  by  a  coil  of  N  turns  of 
insulated  wire  and  each  line  of  induction  links  each  turn  X  =  N<f) 
and  the  total  electromotive  force  induced  in  the  coil  is 

e  =  -N^  abvolts  =  -  10  ~*N  *£  volts  '  (lc) 

dt  dt 

The  minus  sign  in  the  above  equations  is  useful  to  indicate 
the  relative  direction  of  the  induced  electromotive  force  resulting 
from  increasing  the  lines  of  induction  linking  the  circuit  in  the 
same  direction  as  the  lines  of  induction  due  to  the  current  which 
may  be  in  the  circuit.  The  direction  of  the  current  in  a  circuit 
and  the  direction  in  which  the  lines  of  induction  due  to  this  cur- 
rent link  the  circuit  are  always  related  to  each  other  by  the  right- 
handed  screw  law,  while  the  induced  electromotive  force  resulting 
from  an  increase  of  the  number  of  lines  of  induction  linking  the 
circuit  in  the  right-handed  screw  direction  with  respect  to  the 
current  in  the  circuit  is  in  the  opposite  direction  to  the  current. 
In  other  words,  increasing  the  number  of  lines  of  induction  linking 
a  circuit  in  the  same  direction  as  the  lines  of  induction  established 
by  the  current  in  the  circuit,  produces  an  electromotive  force  in 
the  opposite  direction  to  the  direction  of  the  current  ;  that  is, 
produces  a  back  electromotive  force.  A  decrease  in  the  number  of 
lines  of  induction  in  this  same  direction  produces  an  electromotive 
force  in  the  same  direction  as  that  of  the  current.  As  far  as  the 
numerical  value  of  the  electromotive  force  is  concerned,  the  minus 
sign  in  equations  (1)  may  be  neglected. 

106.  Work  Required  to  Change  the  Number  of  Lines  of  Induc- 
tion Linking  an  Electric  Circuit.  —  When  the  electromotive  force 
induced  in  any  circuit  by  changing  the  number  of  linkages  be- 
tween the  flux  and  the  circuit  sets  up  an  electric  current  in  the 
circuit,  or  if  an  electric  current  already  exists  in  the  circuit,  work 
must  be  done  to  produce  this  change  in  the  linkages.  For,  if  the 
current  'has  the  strength  i  and  the  induced  electromotive  force  in 


188  ELECTRICAL  ENGINEERING 

the  direction  of  the  current  the  value  e  at  any  instant,  then,  in 
any  infinitesimal  interval  of  time  dt  measured  from  this  instant, 
the  current  in  this  part  of  the  circuit  gains  an  amount  of  electric 
energy  equal  to  eidt,  where  e  is  the  induced  electromotive  force 
in  the  direction  of  the  current.  But  we  have  just  seen  that  the 
value  of  the  induced  electromotive  force  in  the  direction  of  the 
current  resulting  from  an  increase  d  X  in  the  number  of  linkages 
of  the  flux  and  the  circuit  in  the  same  direction  as  the  lines  of 

induction  due  to  this  current,  is  — — .     Hence  the  amount  of 

dt 

electric  energy  gained  by  the  current  when  the  linkages  of  the 
flux  and  the  circuit  are  increased  by  an  amount  d  \  is 

dW=eidt  =  -  —  idt=  -id\  (2) 

dt 

When  i  is  in  abamperes  and  X  in  maxwells  this  equation  gives  the 
work  done  in  ergs. 

Hence,  when  the  number  of  lines  of  induction  linking  any  part 
of  a  circuit  is  actually  increased,  the  current  in  this  part  of  the 
circuit  loses  electric  energy,  and  when  the  number  of  lines  of 
induction  linking  any  part  of  the  circuit  is  actually  decreased, 
the  current  in  this  part  of  the  circuit  gains  electric  energy.  This 
agrees  with  the  general  principle  which  has  already  been  noted, 
that  an  electromotive  force  in  the  opposite  direction  to  the  direc- 
tion of  the  current,  that  is  a  back  electromotive  force,  corresponds 
to  a  loss  of  electric  energy,  and  an  electromotive  force  in  the  direc- 
tion of  the  current  corresponds  to  a  gain  of  electric  energy. 

107.  Electromotive  Force  Induced  by  the  Cutting  of  Lines  of 
Induction.  —  Right-hand  Rule.  —  When  the  change  in  the  num- 
ber of  lines  of  induction  linking  a  wire  is  caused  by  the  motion  of 
the  wire  through  a  magnetic  field,  the  wire  may  be  looked  upon 
as  cutting  the  lines  of  induction,  and  the  rate  of  change  of  the 
number  of  lines  of  induction  linking  the  wire  may  be  looked  upon 
as  the  rate  at  which  the  wire  cuts  the  lines  of  induction.  The 
direction  of  the  induced  electromotive  force  in  the  wire  will  then 
be  the  direction  in  which  the  middle  finger  of  the  right  hand  points 
when  laid  along  the  wire  and  the  thumb  and  forefinger  of  this 
hand  are  held  perpendicular  to  each  other  and  to  the  middle 
finger,  with  the  thumb  pointing  in  the  direction  of  the  motion 
of  the  wire  and  the  forefinger  in  the  direction  of  the  flux  density. 


ELECTROMAGNETISM 


189 


This  right-hand  rule  for  the  direction  of  the  induced  electromotive 
force  is  equivalent  to  the  rule  given  above,  but  is  more  convenient 
in  the  case  of  a  wire  moving  across  a  magnetic  field.  For  example, 
if  the  lines  of  induction  are  perpendicular  to  the  plane  of  this  page 
and  are  in  the  downward  direction,  a  straight  wire  held  parallel 
to  the  sides  of  the  page  and  moved  from  left  to  right  will  have  an 
electromotive  force  induced  in  it  in  the  direction  from  the  bottom 
to  the  top  of  the  page,  while  if  the  wire  is  moved  from  right  to 
left  the  induced  electromotive  force  will  be  in  the  direction  from 
the  top  to  the  bottom  of  the  page.  In  the  above  equations  for 
the  induced  electromotive  force  (equations  1)  and  the  electric 
energy  gained  by  the  current  (equation  2)  in  any  part  of  a  circuit, 
d  A  may  then  be  interpreted  either  as  the  change  in  the  number 
of  linkages  between  the  flux  and  the  wire  or  as  the  number  of 
lines  of  induction  cut  by  the  wire;  in  the  latter  case  the  electro- 
motive force  is  to  be  taken  as  positive  if  its  direction  as  determined 
by  the  right-hand  rule  is  in  the  direction  of  the  current  in  this 
part  of  the  circuit,  while  if  in  the  opposite  direction  to  that  of  the 
current  it  must  be  taken  as  negative,  i.e.,  a  back  electromotive 
force. 

When  the  change  in  the  number  of  lines  of  induction  linking  a 
wire  is  due  solely  to  the  motion  of  the  wire  through  a  magnetic 
field,  equations  (1)  and  (2)  may  be 
deduced  directly  from  equation  (2) 
of  Chapter  III,  by  the  application 
of  the  principle  of  the  conservation 
of    energy.     Consider    the    special 

case  of  a  straight  wire  of  length  I     ^ 

carrying  an  electric  current  i,  and  F 
let  this  wire  be  in  a  magnetic  field  . 
due  to  any  other  source.  To  sim- 
plify the  discussion  let  the  lines  of 
induction  be  perpendicular  to  the 
wire  and  let  the  flux  density  have 
the  same  value  at  each  element  of 
the  wire.  In  Fig.  66  the  lines  of 
induction  are  taken  in  the  direc- 
tion downward  perpendicular  to  the  plane  of  the  page,  the  wire 
is  taken  parallel  to  the  edge  of  the  page  and  the  direction  of  the 
current  from  the  bottom  to  the  top  of  the  page.  Then  from 


B  Downward 


Motion 


Fig.  66. 


190  ELECTRICAL   ENGINEERING 

equation  (2b)  of  Chapter  III,  the  mechanical  force  produced  on 
the  wire  by  the  agent  producing  the  field  is  F  =  Bli,  and  is  in  the 
direction  from  the  right  to  the  left  (determined  by  the  left-hand 
rule,  see  Article  67).  To  move  the  wire  a  distance  dx  to  the  right 
then  requires  that  some  external  agent  do  an  amount  of  work 
dW=Blidx.  But  Idx  is  the  area  swept  over  by  the  wire  and 
B  is  the  flux  density  normal  to  this  area,  and  since  dx  is  but  an 
infinitesimal  distance,  B  may  be  considered  constant  over  this 
area.  Hence  Bldx  is  the  number  of  lines  of  induction  cut  by 
the  wire  in  moving  the  distance  dx,  that  is  Bldx  =d  <f),  and  there- 
fore the  electric  energy  gained  by  the  current  is  dW  =idl  (f>.  Hence 

the  rate  at  which  the  current  gains  electric  energy  is  i  —  ?  and 

dt 

therefore  the  rise  of  potential  in  the  direction  of  the  current  is 


-     (see  equation  14,  Chapter  III)  which  by  definition  (Article  92) 
dt 

is  equal  to  the  electromotive  force  in  this  direction.  The  numerical 
values  of  the  electromotive  force  and  the  electric  energy  gained 
by  the  current  are  identical  with  those  given  by  equations  (1) 
and  (2).  The  direction  of  the  induced  e.  m.  /.  in  this  case  is  in  the 
direction  of  the  current,  which  agrees  with  the  right-hand  rule 
stated  above.  If  the  wire  is  allowed  to  move  in  the  direction  of 
the  mechanical  force  acting  on  it,  work  is  done  on  the  wire  or 
whatever  opposes  its  motion,  and  the  energy  for  doing  this  work 
comes  from  a  loss  of  electric  energy  by  the  current  in  the  wire. 
In  this  case,  the  induced  electromotive  force  is  in  the  opposite 
direction  to  that  of  the  current,  which  also  agrees  with  the  right- 
hand  rule. 

From  the  relation  —  =BH—  it  also  follows  that  the  value 
dt  dt 

of  the  electromotive  force  in  abvolts  induced  in  the  wire  when  it 
moves  perpendicularly  across  a  magnetic  field  may  also  be  ex- 
pressed by  the  formula 

(3) 


where  v  is  the  velocity  in  centimeters  per  second  at  which  the  wire 
moves  perpendicular  to  itself,  I  is  the  length  of  the  wire  in  centi- 
meters and  B  the  flux  density  in  gausses. 

When  the  wire  moves  through  a  non-magnetic  medium,  the 


ELECTROMAGNETISM 


191 


flux  density  B  at  the  wire  is  equal  to  the  field  intensity  H,  and 
equation  (3)  becomes 


Hlv 


(3a) 


108.  Intensity  of  the  Magnetic  Field  inside  a  Long  Solenoid.  - 

A  useful  application  of  equation  (2)  and  the  conception  of  the 
cutting  of  lines  of  induction  by  a  wire,  is  the  calculation  of  the 
intensity  of  the  magnetic  field  produced  by  an  electric  current 
in  a  coil  made  in  the  form  of  a  long  cylindrical  helix  of  constant 
cross  section;  such  a  coil  is  called  a  solenoid.  Let  Nf  (Fig.  67) 
be  the  number  of  turns  of  wire  per  centimeter  length  of  this  solenoid, 
and  i  the  current  in  abamperes  in  the  wire.  Let  a  unit  north 
point-pole  be  placed  at  any  point  inside  the  solenoid.  There  will 
be  4  TT  lines  of  induction  radiating  out  from  this  unit  point-pole, 
independent  of  the  nature  of  the  medium  inside  the  solenoid, 
whether  it  be  magnetic  or  otherwise  (see  Article  56).  If  the 
diameter  of  the  solenoid  is  small  compared  to  its  length,  practically 
all  the  4  TT  lines  of  induction  which  radiate  out  from  the  unit 
pole  will  pass  through  the  lateral  walls  of  the  solenoid.  Let  the 
pole  be  moved  a  distance  dx  parallel  to  the  axis  of  the  coil ;  then 
each  of  these  4  IT  lines  of  induction  will  move  over  a  distance  dx 
in  the  lateral  surface  of  the  solenoid  and  will  therefore  cut  a  cur- 
rent equal  to  iN'  dx,  provided  the  thickness  of  the  insulation 
between  successive  turns  may  be  neglected.  Since  there  are  4  TT 
of  these  lines  the  total  work  done  against  the  force  produced  on 
the  unit  pole  by  the  current  is  4  TT  iN'  dx,  provided  the  point  at 
which  the  pole  is  placed  is  so  far  from  the  ends  of  the  solenoid  that 


n 

p 

p 

p 

p 

p 

p 

p 

p  t 

)©©©€&©©©©© 

I  x\\  ,--'-""' 

dx              

' 

U 

OOOOOOOOOGi 

Fig.  67. 

the  lines  which  go  out  the  ends  may  be  neglected.  Calling  H 
the  intensity  of  the  magnetic  field  parallel  to  the  axis  of  the  sole- 
noid due  to  the  current  in  it,  the  work  done  against  the  force 
produced  by  the  current  is  also  equal  to  Hdx.  Hence  #  =4  TT  N'i. 
The  relation  between  the  direction  of  the  current  around  the 


192  ELECTRICAL  ENGINEERING 

coil  and  the  direction  of  the  magnetic  field  intensity  through  the 
coil  is  the  same  as  the  relation  between  the  direction  in  which 
a  right-handed  screw  is  turned  and  the  direction  in  which  it 
advances.  The  reader  may  prove  by  a  similar  argument  that  the 
component  of  the  field  intensity  at  any  point  inside  the  solenoid 
at  right  angles  to  its  axis  is  zero,  provided  the  lines  of  induction 
which  go  out  the  ends  may  be  neglected.  (In  proving  this  the 
relative  directions  of  the  lines  of  induction  and  the  current  on  the 
two  sides  of  the  solenoid  must  be  taken  into  account.)  Hence 
the  resultant  field  intensity  in  gilberts  per  centimeter  at  any  point 
inside  a  long  solenoid  a  considerable  distance  from  the  ends  of  the 
solenoid  due  directly  to  the  current  in  the  solenoid  is 

#=4?r  N' i  (4) 

and  is  parallel  to  the  axis  of  the  solenoid,  where  i  is  the  current 
in  abamperes  and  N'  the  number  of  turns  per  centimeter  length. 
Hence  the  magnetic  field  inside  a  long  solenoid  near  its  center 
due  directly  to  the  current  in  the  coil  is  uniform. 

If  the  solenoid  is  wound  on  an  iron  core,  magnetic  poles  wrill 
be  induced  on  the  ends  of  the  core,  and  these  poles  will  also  pro- 
duce a  certain  field  intensity,  which  inside  the  iron  will  be  in  the 
opposite  direction  to  that  due  directly  to  the  current,  but  the 
field  intensity  due  directly  to  the  latter  will  be  exactly  the  same 
as  previously  existed  in  the  air.  When  a  slender  iron  rod,  of  a 
length  considerably  less  than  the  length  of  the  solenoid,  is  placed 
inside  the  solenoid  with  its  axis  parallel  to  the  axis  of  the  solenoid, 
these  induced  poles  may  be  assumed  concentrated  in  points  at 
the  ends  of  the  rod.  Let  the  strength  of  these  poles  be  m  and 
—  m,  and  the  length  of  the  rod  be  /  centimeters.  Then  the  field 
intensity  at  the  center  of  the  rod  due  to  these  induced  poles,  or 
the  so-called  "  demagnetising  force  "  due  to  the  ends  of  the  rod, 
is  approximately 

m  m          8m 

ft" 


&  (4) 


and  the  resultant  field  intensity  at  the  center  of  the  rod  is  therefore 
Hr=H-  Ht=4irN'i — j  (4«) 

An  exact  determination  of  the  pole  strength  m  is  extremely 
difficult,  and  for  this  reason  magnetic  tests  are  seldom  made  on 


ELECTROMAGNETISM  1  93 

rods;    instead  a  ring  or  rod  and  yoke  are  usually  employed  (see 
Article  109). 

109.  Determination  of  the  Number  of  Lines  of  Induction  Link- 
ing an  Electric  Circuit.  —  Measurement  of  Quantity  of  Elec- 
tricity. —  From  the  fact  that  an  electromotive  force  is  induced  in  an 
electric  circuit  when  the  number  of  lines  of  induction  threading  the 
circuit  is  changed,  it  is  possible  by  a  simple  experiment  to  measure 
the  number  of  lines  of  induction  through  any  region  in  space. 
Consider  first  a  coil  of  TV  turns  connected  in  a  circuit  of  which 
the  total  resistance,  including  that  of  the  coil,  is  R,  and  let  the 
number  of  lines  of  induction  threading  this  circuit  be  changed 
from  <£,  to  (j>2,  and  let  t  be  the  time  required  for  this  change  to 
take  place.  Then  the  average  e.  m.  f.  induced  in  the  circuit 

during  this  interval  is  numerically  E  =LL±  —  rll  —  provided  each 

t 

line  of  induction  links  each  turn,  and  therefore  the  average  value 
of  the  induced  current  is 


R  Rt 

or  (,-< 


R 

But  It  is  the  quantity  of  electricity  which  flows  through  the 
circuit  in  this  interval  t.  Hence  the  important  relation  that, 
when  the  number  of  lines  of  induction  linking  a  coil  of  N  turns 
is  changed  by  an  amount  (j>l  —  <f>2,  a  quantity  of  electricity 

0_(<f>1-<f>2)N  (5) 

R 

is  transferred  across  each  section  of  the  wire  forming  the  circuit, 
where  R  is  the  total  resistance  of  the  circuit,  where  all  quantities 
are  in  c.  g.  s.  units.  Hence,  if  when  a  coil  which  forms  part  of  a 
circuit  which  has  a  total  resistance  of  R  abohms  is  pulled  quickly 
out  of  a  magnetic  field,  a  quantity  of  Q  abcoulombs  of  electricity 
is  "  discharged  "  through  the  circuit,  the  number  of  lines  of  in- 
duction in  maxwells  which  threaded  the  coil  when  it  was  in  the 
field  is 

,     R  Q 


N 
Or,  if  the  coil  is  reversed  in  the  field  (i.e.,  turned  180  degrees  about 


194  ELECTRICAL   ENGINEERING 

any  axis  in  the  plane  of  the  coil),  or  if  the  direction  of  the  field 
through  the  coil  is  reversed,  the  number  of  lines  of  induction  in 
maxwells  originally  threading  the  coil  is 

*- 


When  a  momentary  current  is  established  in  a  galvanometer 
of  which  the  moving  element  is  fairly  heavy  and  has  a  long  period 
of  vibration,  it  can  be  shown  that  the  momentary  force  or  impulse 
produced  on  this  element  causes  it  to  swing  out  from  its  position 
of  equilibrium  by  an  amount  which  is  approximately  proportional 
to  the  quantity  of  electricity  discharged  through  the  galvanom- 
eter.* Hence,  if  the  coil  we  have  just  been  considering  is  con- 
nected in  series  with  such  a  long  period  or  ballistic  galvanometer, 
and  the  change  in  the  number  of  lines  of  induction  threading  the 
coil  is  made  so  rapidly  that  the  current  established  in  the  circuit 
lasts  for  only  a  small  fraction  of  the  time  required  for  the  moving 
element  of  the  galvanometer  to  make  a  complete  swing,  the  first 
swing  of  the  moving  element  will  be  proportional  (approximately) 
to  the  quantity  of  electricity  Q  discharged  through  the  circuit, 

that  is,  proportional  to  —  ?-  —  _L!L_ 

R 

From  the  relation  established  in  the  preceding  article,  the 
change  in  the  number  of  lines  of  induction,  <^>1—  *<£2,  can  be  calcu- 
lated in  the  special  case  when  the  coil  is  wound  on  a  non-magnetic 
spool  which  is  placed  inside  and  at  the  center  of  a  long  solenoid 
with  an  air  core,  and  the  current  in  the  solenoid  is  reversed  in 
direction.  Let 

N\  =the  number  of  turns  per  centimeter  length  of  the  solenoid 

or  "  primary  "  coil. 
7=the  strength  of  the  current  in  abamperes  in  the  solenoid 

or  "  primary  "  coil. 

JV2=the  number  of  turns  on  the  spool  or  "  secondary  "  coil. 
R=the  total  resistance  in  abohms  of  the  secondary  coil,  the 
galvanometer  and  any  extra  resistance  which  may  be 
in  the  secondary  circuit. 
*S=the  area  in  square  centimeters  of  the  mean  cross  section 

of  the  secondary  coil. 

Then  the  field  intensity  at  each  point  of  the  area  S  is  H  =4  TT  N\  I, 
*J.  J.  Thomson,  Elements  of  Electricity  and  Magnetism,  p.  377. 


ELECTROMAGNETISM  195 

which  is  also  equal  to  the  flux  density,  since  the  permeability 
is  unity.  Hence  the  number  of  lines  of  induction  threading  the 
secondary  coil  is  4  TT  N'JS.  Hence,  when  the  current  in  the 
primary  coil  is  reversed,  the  change  in  the  number  of  lines  of 
induction  threading  the  secondary  coil  is  8  TT  N'JS.  The  quan- 
tity of  electricity  discharged  through  the  galvanometer  is  then 

87rN'JSN2  (6; 

R 

where  all  quantities  are  in  c.  g.  s.  units.  Hence,  by  calculating  Q 
and  noting  the  galvanometer  swings  when  currents  of  various 
strengths  are  reversed  in  the  primary  coil,  a  curve  can  be  plotted 
giving  the  quantity  of  electricity  corresponding  to  a  swing  of  any 
value;  such  a  curve  will  be  approximately  a  straight  line.  We 
have  then  an  instrument  for  measuring  the  quantity  of  electricity 
t  -ansf erred  through  a  circuit  by  a  momentary  current,  and  con- 
sequently a  means  of  determining  the  quantity  Q  in  equations 
(4),  and  therefore  a  means  for  measuring  the  change  in  the  number 
of  lines  of  induction  linking  any  coil. 

110.  Determination  of  the  B  — H  Curve  and  Hysteresis  Loop.  - 
A  useful  application  of  equations  (5)  is  the  determination  of  the 
B—  H  curve  and  hysteresis  loop  (see  Article  57)  of  a  sample  of 
iron  or  other  magnetic  substance  made  in  the  form  of  a  closed 
"  anchor "  ring  or  "  toroid."  Let  such  a  ring  be  uniformly 
wound  with  a  coil  of  wire  C'  which  is  connected  through  a  suitable 
switch  S  and  a  variable  resistance  or  rheostat  Rf  to  a  battery  B 
or  other  source  of  electromotive  force.  Over  this  primary  wind- 
ing on  the  iron  ring  let  a  second  coil  C  be  wound  and  connected 
through  a  resistance  R"  to  a  ballistic  galvanometer  G  which  has 
been  calibrated  in  the  manner  described  in  the  preceding  article. 
Let  NI  be  the  total  number  of  turns  of  the  primary  coil,  Na  the 
total  number  of  turns  in  the  secondary  coil,  and  R  the  total 
resistance  of  the  secondary  coil,  the  galvanometer  and  the  resist- 
ance R"  in  series  with  the  coil  and  galvanometer.  (The  resistance 
R"  is  inserted  in  the  secondary  circuit  so  that  the  galvanometer 
deflection  may  be  kept  within  the  range  of  the  galvanometer  scale.) 
It  can  be  shown  (Article  114)  that  when  a  current  of  I  abamperes 
is  established  in  the  primary  coil  the  lines  of  force  set  up  in  the 
ring  are  circles  concentric  with  the  center  of  the  ring  and  the 

average  intensity  of  this  magnetic  field  is   H  =  -  -  gilberts 


196 


ELECTRICAL  ENGINEERING 


per  centimeter,  where  /  is  the  length  in  centimeters  of  the  mean 
circumference  of  the  ring.  (This  formula  is  an  approximation, 
and  applies  only  when  the  radial  thickness  of  the  material  forming 
the  ring  is  small  compared  with  the  radius  of  the  mean  circum- 
ference of  the  ring,  i.e.,  the  dotted  line  in  Fig.  68.)  A  circular 


Mean  circumference 


Fig.  68. 

ring  thus  uniformly  wound  with  a  coil  of  wire  carrying  an  electric 
current  has  no  magnetic  poles  induced  on  it,  hence  this  value  of 
H  is  the  total  field  intensity  inside  the  ring  and  is  independent  of 
the  magnetic  nature  of  the  ring.  When  the  ring  is  made  of  a 
magnetic  material  such  as  iron,  the  flux  density,  i.e.,  the  number 
of  lines  of  induction  per  square  centimeter,  inside  the  ring  will 
not  be  equal  to  the  field  intensity  H  but  will  have  some  other 
value  B.  These  lines  of  induction,  however,  will  coincide  in 
direction  with  the  lines  of  force,  and  therefore  across  any  radial 
section  A  of  the  ring  there  will  be  BA  lines  of  induction.  Hence, 
if  the  ring  is  originally  unmagnetised*  and  a  current  of  7  ab am- 
peres is  established  in  the  primary  winding,  the  number  of  lines 
of  induction  through  the  secondary  coil  will  change  by  an  amount 
<f>=BA  as  the  result  of  establishing  in  the  ring  a  field  intensity 

H  =—        ^.    This  change  of  BA  lines  of  induction  through  the 

coil  will  cause  a  transfer  of    Q  =  —       —  abcoulombs  of  electricity 

R 

*The  ring  can  be  demagnetised  at  the  start  by  reversing  the  current  in  the 
primary  winding  back  and  forth  and  gradually  decreasing  its  strength. 


ELECTROMAGNETISM  197 

through  the  galvanometer,  producing  a  swing  of  the  moving  ele- 
ment. The  quantity  of  electricity  Q  corresponding  to  this  swing 
can  be  read  directly  from  the  calibration  curve  of  the  instrument, 
and  therefore  the  flux  density  B  corresponding  to  the  field  in- 

7?  (~) 

tensitv  H  can  be  calculated  from  the  formula  B= -. 

AN2 

By  increasing  the  value  of  the  primary  current  in  successive 
steps  and  noting  the  deflection  of  the  galvanometer  corresponding 
to  each  change  in  the  primary  current,  the  B—  H  curve  for  the 
sample  is  readily  determined.  To  determine  the  hysteresis  loop 
corresponding  to  any  maximum  field  intensity  H,  the  primary 
current  is  reversed  back  and  forth  a  number  of  times  between 
the  positive  and  negative  values  corresponding  to  H  and  —  H, 
and  is  then  decreased  in  steps  from  the  value  corresponding  to  H 
to  the  value  corresponding  to  —  H,  and  the  galvanometer  deflec- 
tion corresponding  to  each  step  noted.  The  primary  current  is 
then  increased  in  steps  from  —  H  to  H,  and  the  corresponding 
deflections  of  the  galvanometer  again  noted.  From  these  obser- 
vations the  value  of  the  flux  density  corresponding  to  each  value 
of  H  in  this  cycle  of  changes  may  be  calculated,  and  hence  the 
hysteresis  loop  may  be  plotted. 

111.  The  Continuous  Current  Dynamo. —  The  name  continu- 
ous current  dynamo  is  given  to  any  machine  in  which  a  constant 
or  continuous  electromotive  force  is  developed  by  the  rotation 
of  one  or  more  conductors  in  a  magnetic  field.  Such  a  machine 
may  be  used  as  a  generator  of  electric  energy  when  driven  by 
some  external  means  such  as  a  steam  engine,  water  wheel  or  gas 
engine ;  or  when  electric  energy  is  supplied  to  it,  it  may  be  used  as 
an  electric  motor,  converting  electric  energy  into  mechanical 
energy.  The  principles  involved  in  the  construction  of  dynamo, 
whether  it  is  designed  for  use  as  a  generator  or  as  a  motor,  are 
identical. 

The  conductors  in  which  the  electromotive  force  is  induced 
in  a  continuous  current  generator  are  usually  insulated  copper 
wires  or  copper  bars,  called  armature  conductors,  which  are  im- 
bedded in  slots  in  the  exterior  surface  of  a  hollow  iron  cylinder 
(see  Fig.  69).  These  slots  are  parallel  to  the  axis  of  the  cylinder. 
The  cylinder  itself,  which  is  called  the  armature  core,  is  made  up 
of  sheets  of  soft  iron  or  sheet  steel  which  has  a  high  permeability  ; 
the  planes  of  these  sheets  are  perpendicular  to  the  axis  of  the 


198 


ELECTRICAL  ENGINEERING 


cylinder.  In  large  machines  these  flat  rings  of  sheet  steel  are 
fitted  on  to  a  cast  iron  frame,  called  the  armature  spider,  which 
resembles  the  spokes  and  hub  of  a  wheel.  The  armature  spider 
(or,  in  small  machines,  the  steel  sheets  themselves)  is  mounted  on 
a  shaft  or  axle  which  runs  through  it  and  projects  out  at  each  end. 
The  ends  of  the  shaft  are  mounted  in  suitable  bearings,  and,  in 
case  the  machine  is  to  be  driven  by,  or  is  to  drive,  a  belt,  a  pulley 
is  mounted  on  one  end  of  the  shaft.  The  armature  conductors, 
core  and  spider  taken  collectively  are  called  the  armature  of  the 
machine. 

The  magnetic   field   in  which   the   armature   conductors   are 
rotated  is  produced  by  an  electric  current  in  two  or  more  coils 


Armature  Core 
Armature  Conductors 
Shaft 


Average  Path 
Lines  of  Induction 

Fig.  69. 

of  insulated  wire,  called  field  coils,  which  are  wound  on  stationary 
iron  or  steel  cores  which  are  placed  symmetrically  around  the 
armature  core,  as  shown  in  the  figure.  The  ends  of  these  field 
cores  next  the  armature  are  broadened  out  so  that  they  cover 
from  50  to  70  per  cent  of  the  armature  surface,  and  are  made 
concave  toward  the  armature  so  that  the  end  surfaces,  called 
the  pole  faces,  form  part  of  a  cylindrical  surface  concentric  with 


ELECTROMAGNETISM  199 

the  armature  and  of  a  slightly  greater  radius.  These  broadened 
ends  of  the  field  cores  are  called  the  pole  shoes,  and  are  frequently 
made  separate  from  the  field  cores  and  are  bolted  to  the  latter 
when  the  machine  is  assembled.  The  air  space  between  the  pole 
shoes  and  the  armature  core  is  called  the  air  gap.  The  ends  of 
the  field  cores  away  from  the  armature  are  connected  by  a  yoke 
of  iron  or  steel ;  this  yoke  is  called  the  field  yoke.  The  field  coils, 
field  cores,  pole  shoes  and  field  yoke  taken  collectively  are  called 
the  field  of  the  machine.  The  field  cores,  pole  shoes,  air  gap, 
field  yoke,  and  armature  core  taken  collectively  are  called  the 
magnetic  circuit  of  the  machine,  since  the  lines  of  induction  are 
practically  all  confined  to  the  space  occupied  by  these  parts. 
The  average  permeability  of  this  magnetic  circuit  is  high,  and 
consequently  a  comparatively  small  amount  of  energy  is  required 
to  maintain  an  electric  current  of  sufficient  strength  in  the  field 
coils  to  establish  a  large  number  of  lines  of  induction  through 
the  armature  winding.  (See  Article  113.)  The  field  cores,  pole 
shoes  and  field  yoke  form  a  powerful  electromagnet  when  an 
electric  current  is  established  in  the  field  coils ;  the  lines  of 
force  enter  the  air  gap  from  the  north  pole  of  this  magnet,  and 
pass  from  the  air  gap  into  the  south  pole  of  this  magnet. 

In  the  simplest  type  of  armature  winding,  the  armature 
conductors  are  all  connected  in  series  by  insulated  copper  wires 
or  bars,  called  the  end  connectors,  across  the  two  ends  of  the 
armature  core,  and  form  a  closed  coil  around  this  core.  Hence 
the  net  electromotive  force  induced  in  this  closed  winding  is  the 
sum  of  the  electromotive  forces  induced  in  the  individual  arma- 
ture conductors.  The  field  winding  produces  practically  no  lines 
of  induction  which  cut  the  end  connectors ;  the  entire  electromotive 
force  induced  in  the  armature  winding  is  that  due  to  the  cutting 
of  lines  of  induction  by  the  armature  conductors  which  lie  in  the 
air  gap.  Hence,  when  the  armature  conductors  are  distributed 
symmetrically  around  the  armature  core,  the  net  electromotive 
force  induced  in  the  armature  winding  is  zero,  since  for  each 
conductor  cutting  the  magnetic  field  on  one  side  of  the  air  gap, 
there  is  a  corresponding  conductor  cutting  an  equal  field  on  the 
other  side  of  the  armature  in  the  opposite  direction  with  respect 
to  the  motion  of  the  conductor.  A  study  of  Fig.  69  will  make 
this  clear;  a  dot  in  an  armature  conducto'r  indicates  an  electro- 
motive force  in  the  direction,  toward  the  reader,  and  a  cross  an 


200 


ELECTRICAL   ENGINEERING 


electromotive  force  in  the  direction  away  from  the  reader;  the 
e.  m.  f.  induced  in  those  conductors  not  under  a  field  pole  is  practi- 
cally zero.  The  average  paths  of  the  lines  of  induction  are 
shown  by  the  dotted  lines. 

In  spite  of  the  fact  that  the  net  electromotive  force  induced 
in  the  closed  armature  winding  is  zero,  a  continuous  electromotive 
force  can  be  obtained  from  this  machine  by  bringing  out  suitable 
connections  or  taps  from  the  winding,  and  making  contact  with 
these  connections  in  a  suitable  manner.  This  can  be  best  under- 
stood from  Fig.  70,  which  shows  diagrammatically  an  armature 
winding  of  a  simple  two-pole  machine,  with  the  end  connectors  on 
the  end  of  the  armature  facing  the  reader  in  heavy  lines  and  the  end 
connectors  on  the  other  end  as  dotted  lines.  The  middle  point  of 
each  of  the  front  connections  is  connected  to  a  bar  of  a  device  called 
a  commutator,  which  is  a  cylinder  of  copper  bars  insulated  from 
each  other  by  sheets  of  mica.  This  cylinder  is  mounted  rigidly 
on  the  shaft  of  the  armature,  from  which  the  bars  are  also  insulated. 
From  the  segment  marked  1  there  are  two  paths  through  the  arma- 
ture winding  to  the  segment 
marked  5,  and  the  electro- 
motive force  induced  in  each 
of  the  armature  conductors 
in  each  of  these  paths  is  in 
the  direction  from  1  to  5. 
Hence  the  net  electromotive 
force  between  1  and  5 
through  each  of  these  paths 
is  the  arithmetical  sum  of 
the  electromotive  forces  in- 
duced in  all  the  conductors 
in  each  path,  i.e.,  in  half  the 
total  number  of  armature 
conductors.  The  two  halves 
of  the  armature  winding 
are  then  similar  to  two  electric  batteries  in  parallel;  the  electro- 
motive force  of  one  half  the  winding  opposes  the  electromotive 
force  induced  in  the  other  half  of  the  winding  and  consequently 
no  current  flows  in  the  closed  circuit  formed  by  the  entire  winding. 
However,  when  the  segments  1  and  5  are  connected  by  an  external 
conductor,  the  electromotive  force  impressed  on  this  circuit  will  be 


ELECTROMAGNETISM  201 

equal  to  the  electromotive  force  induced  in  each  half  of  the  wind- 
ing, and  consequently  a  current  will  be  established  in  the  external 
conductor,  one  half  of  the  current  flowing  through  each  half  of 
the  armature  winding;  just  as  in  the  case  of  two  batteries  in 
parallel  and  connected  to  an  external  circuit  half  the  current 
flows  through  each  battery,  provided  the  electromotive  forces 
and  the  internal  resistances  of  the  two  batteries  are  respectively 
equal.  As  the  armature  turns  in  the  direction  indicated  the 
electromotive  forces  induced  in  the  individual  conductors  form- 
ing each  path  between  1  and  5  will  not  all  be  in  the  same  direction, 
but  some  of  the  electromotive  forces  will  oppose  the  other,  and 
hence  the  net  electromotive  force  between  1  and  5  will  decrease. 
However,  when  the  armature  has  rotated  through  an  angle 
corresponding  to  one  segment  of  the  commutator,  another  pair 
of  segments,  8  and  4,  come  into  the  position  formerly  occupied 
by  1  and  5  and  the  electromotive  force  between  8  and  4  will  be 
exactly  the  same  as  the  electromotive  force  which  previously 
existed  between  1  and  5,  provided  the  armature  rotates  with 
a  constant  speed.  Similarly  for  the  next  pair  of  segments,  and 
so  on.  Consequently,  if  in  the  positions  B  and  Bf  are  mounted 
two  fixed  contacts,  under  which  the  commutator  segments  slide 
as  the  armature  rotates,  the  electromotive  force  between  these 
two  contacts  will  remain  practically  constant,  and  the  greater 
the  number  of  commutator  segments  the  more  nearly  constant 
will  this  electromotive  force  be.  The  fixed  contacts  which  rub 
against  the  commutator  segments  are  called  brushes,  and  in  most 
modern  machines  are  made  of  carbon  blocks.  These  brushes  are 
held  in  a  suitable  support,  called  the  brush-holder,  and  the  parts 
of  this  brush-holder  in  contact  with  the  brushes  are  insulated  by 
suitable  bushings  from  the  brackets  which  connect  them  to  a 
common  support  called  the  rocker-arm,  which  is  mounted  on 
some  part  of  the  stationary  structure  which  forms  the  frame 
of  the  machine.  This  rocker-arm  is  so  mounted  that  the  position 
of  the  brushes  can  be  adjusted  until  no  sparking  occurs  between 
them  and  the  commutator  segments  when  a  current  is  established 
through  the  machine.  The  brush  at  the  lower  electric  potential 
is  called  the  negative  brush,  and  the  brush  at  the  higher  electric 
potential  is  called  the  positive  brush.  The  electromotive  force 
induced  in  a  dynamo  is  always  from  the  negative  to  the  positive 
brush,  whether  the  dynamo  be  used  as  a  generator  or  as  a  motor. 


202 


ELECTRICAL   ENGINEERING 


Series  Connected 


When  the  machine  is  to  be  used  as  a  generator  it  is  driven 
by  some  form  of  "  prime  mover,"  i.e.,  a  steam  engine,  gas  engine, 
or  water  wheel,  and  the  circuit  which  is  to  be  supplied  with 
electric  energy  is  connected  in  series  with  the  two  brushes  B  and 
B'.  The  field  coils  may  be  connected  either  in  series  with  this 
external  circuit;  or  they  may  be  connected  directly  across  the 
brushes  of  the  machine;  or  there  may  be  two  sets  of  field  coils, 
one  set  connected  across  the  brushes  and  the  other  set  in  series 
with  the  external  circuit.  In  the  first  case  the  machine  is  called 
a  series  connected  generator,  in  the  second  a  shunt  connected 
generator,  and  in  the  third  case  a  compound  connected  generator. 
These  various  forms  of  connections  are  shown  in  Fig.  71.  All 
these  types  of  generators  are  called  self-excited  generators,  since 

they  produce  their  own  magnetic  field. 
There  is  in  general  sufficient  "  resid- 
ual magnetism  "  in  the  iron  part  of 
the  machine  to  establish  a  weak  mag- 
netic field  in  the  air  gap,  which  in  turn 
establishes  a  small  electromotive  force 
between  the  brushes  when  the  arma- 
ture is  rotated ;  this  electromotive  force 
in  turn  establishes  a  small  current  in 
the  field  windings  which  increases  the 
magnetic  field  in  the  gap ;  this  in  turn 
increases  the  induced  electromotive 
force,  and  this  cumulative  process  goes 
on  until  the  field  current  reaches  a 
steady  value.  In  the  simple  shunt 
connected  generator,  for  example,  the 
field  current  increases  until  the  in- 
duced electromotive  force  in  the  arma- 
ture establishes  a  difference  of  potential 
between  the  brushes  of  the  machine 
^r^r>r^\_  equal  to  the  product  of  the  resistance 
of  the  shunt  field  by  the  strength  of 
the  current  in  this  field. 
~ ^  The  total  current  taken  from  the 

Compound  Connected  i  i  />  c  ,  • 

Fig.  71  brushes  of  any  type  of  continuous  cur- 

rent generator  for  a  given  induced,  or 
"  armature,"   electromotive  force,  will  depend  upon  the  resist- 


Shunt  Connected 


Series  Field 


ELECTROMAGNETISM  203 

ances  and  electromotive  forces  in  the  external  circuit  and  also 
upon  the  resistances  of  the  armature  circuit  between  the  brushes 
and  upon  the  resistances  of  the  field  windings.  When  the  various 
resistances  and  electromotive  forces  are  known  the  strengths  of 
the  currents  in  the  external  circuit  and  the  various  windings  of 
the  machine  can  be  calculated  by  Kirchhoff's  Laws.  Since  in  an 
electric  generator  the  current  always  gains  electric  energy,  the 
direction  of  the  current  through  the  armature  is  the  same  as  that 
of  the  electromotive  force  developed  in  the  armature.  Hence  the 
current  always  leaves  the  armature  of  a  generator  at  the  positive 
brush  and  enters  at  the  negative  brush. 

A  continuous  current  dynamo,  series,  shunt  or  compound 
connected,  may  also  be  used  as  a  motor.  In  this  case  its  ter- 
minals are  connected  to  some  source  of  potential  difference  which 
establishes  a  current  through  its  field  coils  and  through  its  arma- 
ture. The  magnetic  field  produced  by  the  current  in  the  field 
coils  exerts  a  mechanical  force  on  the  armature  and  causes  it  to 
rotate.  Work  is  then  done  on  whatever  is  connected  to  the 
armature,  and  the  energy  to  do  this  work  comes  from  the  source 
of  electromotive  force,  e.g.,  the  generator,  which  establishes  the 
current  through  the  machine.  Since  to  transfer  the  energy  from 
the  generator  to  the  motor  it  is  only  necessary  to  have  two  wires 
or  mains,  as  they  are  usually  called,  the  motor  may  be  at  a  great 
distance  from  the  generator ;  hence  the  great  advantage  of  trans- 
mitting energy  by  means  of  an  electric  current.  Since  in  an 
electric  motor  electric  energy  is  always  lost  by  the  current  in  the 
motor,  the  electromotive  force  developed  by  a  motor  is  always 
in  the  direction  opposite  to  that  of  the  current ;  that  is,  an  electric 
motor  always  develops  a  back  electromotive  force.  The  current 
therefore  always  enters  the  armature  of  a  motor  at  the  positive 
brush  and  leaves  it  at  the  negative  brush. 

For  a  fuller  description  of  the  construction  of  electric  dynamos 
and  a  discussion  of  the  various  factors  which  affect  their  operation 
as  generators  or  motors,  the  reader  is  referred  to  any  text-book 
on  dynamo-electric  machinery.  In  particular,  it  should  be  noted 
that  dynamos  designed  for  the  generation  or  utilisation  of  large 
amounts  of  electric  power  in  general  have  a  number  of  pairs 
of  poles  and  in  most  cases  a  corresponding  number  of  sets  of 
brushes,  with  all  the  positive  brushes  interconnected  and  all  the 
negative  brushes  interconnected.  The  armature  in  such  "  multi- 


204  ELECTRICAL  ENGINEERING 

polar  "  machines  may  have  a  single  winding  of  the  form  described 
above,  or  there  may  be  two  or  more  independent  windings  on 
the  same  armature  core.  The  general  principles  involved  in 
the  construction  and  operation  of  such  machines  are,  however, 
identically  the  same  as  in  the  case  of  the  simple  two-pole  machine. 
112.  Calculation  of  the  Electromotive  Force  Induced  in  the 
Armature  of  a  Continuous  Current  Dynamo.  —  The  value  of  the 
electromotive  force  induced  in  the  armature  winding  of  a  continu- 
ous current  dynamo  may  be  readily  calculated  from  equation  ( 1) . 
Let 

N  =the  total  number  of  armature  conductors. 
p  =the  number  of  field  poles. 
p'  =the  number  of  parallel  conducting  paths   between  the 

JV 

positive  and  negative  brush  sets,  that  is,  —  is  equal  to 

p 

the  number  of  armature  conductors  in  series  between 
the  positive  and  negative  brush  sets. 

<f)  =the  total  useful  magnetic  flux  per  pole  in  c.  g.  s.  lines  or 
maxwells ;  that  is,  <£  is  the  total  number  of  lines  of  in- 
duction which  pass  through  the  armature  core  from  a 
north  pole  to  the  adjacent  south  poles  of  the  field  magnet. 

n  =the  number  of  revolutions  of  the  armature  per  second. 
The  time  required  for  each  conductor  to  pass  entirely  around 

the  armature  is  then  —  and  therefore  the  time  taken  for  it  to  pass 
n 

through  the  magnetic  field  under  each  pole  is   — .     Hence  the 

up 

average  value  of  the  electromotive  force  induced  in  each  armature 
conductor  as  it  passes  under  each  pole  is  np  <f>  abvolts.  Since 
the  conductors  are  uniformly  distributed  around  the  surface  of 
the  armature,  this  is  also  the  average  value  at  each  instant  of 
the  electromotive  force  induced  in  these  conductors.  Since 

N 
there  are  —  conductors  in  series  between  the  positive  and  negative 

P' 

brush  sets,  the  average  value  of  the  total  electromotive  force 
between  the  brushes  is 

„    np6N      ..  (7) 

E  =  — L. —  volts 

p'XlO8 
When   the   number   of   commutator  segments   is   large,   the   in- 


ELECTROMAGNETISM  205 

stantaneous  values  of  the  electromotive  force  between  the  brushes 
are  practically  equal  to  this  average  value.  For,  since  the  brushes 
always  make  contact  with  segments  in  the  same  position  with 
respect  to  the  field  which  produces  the  flux,  the  only  possible 
variation  in  the  electromotive  force  between  the  brushes  (when 
(f>  and  n  remain  constant)  is  the  variation  that  might  occur  as 
the  armature  rotates  through  an  angle  corresponding  to  one 
commutator  segment.  But  in  the  distance  corresponding  to 
this  small  displacement  of  an  armature  conductor,  the  flux  density 
remains  practically  constant  (except  for  the  three  or  four  con- 
ductors which  are  just  passing  under  or  are  just  leaving  a  pole 
tip),  and  hence  the  rate  at  which  the  conductors  are  cutting 
lines  of  induction,  and  therefore  the  induced  electromotive  force, 
does  not  change  appreciably  in  this  interval. 

Equation  (7)  holds  whether  the  dynamo  is  used  as  a  generator 
or  as  a  motor.  In  the  case  of  a  generator,  this  electromotive 
force  is  in  the  direction  of  the  current,  and  in  case  of  a  motor 
in  the  opposite  direction  from  that  of  the  current.  It  should 
be  noted,  however,  that  the  flux  per  pole  in  a  generator  or  motor 
in  general  depends  not  only  upon  the  current  in  the  field  coils 
but  also  upon  the  current  in  the  armature ;  this  is  caused  by  the 
fact  that  the  armature  current  also  sets  up  lines  of  induction  in 
the  opposite  direction  to  those  due  to  the  field  current  and  the 
resultant  flux  per  pole  is  decreased.  See  under  Armature  Re- 
action in  any  text-book  on  dynamo-electric  machinery. 

113.  Magnetomotive  Force.  — In  equation  (7)  for  the  electro- 
motive force  of  a  continuous  current  dynamo,  the  only  quantity 
which  cannot  be  readily  predetermined  is  the  flux  per  pole.  This 
quantity  can,  however,  be  calculated  when  the  dimensions  of 
the  magnetic  circuit,  the  permeability  of  its  various  parts,  the 
number  of  turns  on  the  field  coils  and  the  "back  ampere-turns" 
of  the  armature  are  known.  The  following  considerations  will 
make  this  clear. 

Consider  a  wire  wound  into  a  coil  of  any  form  having  N  turns, 
and  let  a  current  of  /  abamperes  be  established  in  this  wire. 
We  have  already  seen  that  a  magnetic  field  will  be  established 
around  such  a  coil,  and  that  the  lines  of  force  representing  this 
field  will  be  closed  loops  linking  the  coil.  Consequently,  between 
any  two  points  in  the  field  around  this  coil  there  will  be  in  general 
a  drop  (or  rise)  of  magnetic  potential.  We  wish  to  find  the 


206  ELECTRICAL  ENGINEERING 

relation  between  the  total  drop  of  potential  around  any  closed 
path  linking  this  coil  and  the  value  of  the  current  established 
in  it.  By  definition  (Article  60),  the  drop  of  magnetic  potential 
along  any  path  in  a  magnetic  field  is  the  work  done  by  the  agent 
producing  the  field  when  a  unit  north  point-pole  moves  around 
this  path,  that  is,  the  drop  of  potential  around  any  closed  path 

is  the  line  integral  J      (H  cos  0)  dl  around  that  path,  where  dl  is 

any  elementary  length  in  the  path,  H  the  field  intensity  at  dl,  0 
the  angle  between  the  direction  of  dl  and  the  direction  of  H,  and 

the  symbol  f  L|  represents  the  integral  around  this  closed  path. 

We  have  also  seen  that  it  is  physically  impossible  to  have  a 
north  magnetic  pole  without  at  the  same  time  having  an  equal 
south  magnetic  pole  on  the  same  piece  of  matter.  Hence  it  is 
physically  impossible  to  move  a  unit  north  pole  around  a  closed 
path  linking  a  coil  without  at  the  same  time  linking  the  coil 
with  an  equal  south  pole,  unless  the  magnet  at  the  ends  of  which 
these  poles  exist  is  threaded  through  and  bent  into  a  closed  loop 
linking  the  coil.  Hence  to  move  a  unit  north  point-pole  around 
a  closed  coil  without  at  the  same  time  having  the  current  in  the 
coil  exert  a  force  on  the  south  pole  connected  to  this  north  pole, 

©  we  may  conceive  of  this  unit 

north  point-pole  as  at  the  end 

original  position  of  a  flexible  magnetic  filament, 

the  south  pole  of  which  is  so  far 
removed  from  the  coil  that  the 
force  exerted  by  the  latter  on 
I  this  south  pole  is  negligible. 
'The  north  pole  end  of  this  fila- 
ment can  then  be  threaded 
through  the  coil  and  back 
Flg-  72>  around  to  its  original  position 

over  the  desired  path,  as  shown  in  Fig.  72.  The  only  force 
exerted  by  the  coil  on  the  filament  as  it  is  thus  bent  around 
will  be  the  force  produced  by  the  coil  on  the  north  pole  of 
the  filament,  and  therefore  the  work  done  by  the  current  will 

be  J  (H cos  9)dl.  But  we  have  also  seen,  Article  56,  that  4  TT 
lines  of  induction  exist  in  a  filament  which  has  unit  poles, 
in  the  direction  from  the  south  to  the  north  pole  of  the  fila- 


ELECTROMAGNETISM  207 

ment.  Hence,  when  the  unit  north  pole  is  moved  through  the 
coil  in  the  right-handed  screw  direction  with  respect  to  the  current 
and  back  again  along  a  closed  path  linking  the  coil,  the  number 
of  lines  of  induction  through  the  coil  is  increased  by  4-7T.  Hence 
from  equation  (2),  the  work  done  by  the  current  on  the  pole  is 
also  equal  to  4?r  NI,  since  the  total  number  of  linkages  is  ^irN. 
Hence  the  drop  of  magnetic  potential  around  a  closed  path  linking 
a  coil  of  N  turns  in  the  right-handed  screw  direction  with  respect 
to  the  current  in  the  coil  is  equal  to  4ir  N  times  the  strength  of  the 
current  in  the  wire  forming  the  coil.  That  is 

(HcosO)dl=4:TTNI  (8) 

This  relation  is  analogous  to  that  between  the  resistance  drop  in 
an  electric  circuit  and  the  electromotive  force  in  the  circuit,  see 
equation  (25c)  of  Chapter  III.  Hence  the  expression  4:irNI  is 
called  the  magnetomotive  force  of  the  coil. 

Magnetomotive  force  is  measured  in  the  same  unit  as  drop 
of  magnetic  potential,  that  is,  the  c.  g.  s.  electromagnetic  unit  of 
magnetomotive  force  is  the  gilbert.  The  magnetomotive  force  is 
proportional  to  the  product  of  the  number  of  amperes  in  each 
turn  of  the  coil  and  to  the  number  of  turns  in  the  coil;  this 
product  NI,  when  I  is  expressed  in  amperes,  is  called  the  ampere- 
turns  of  the  coil.  The  magnetomotive  force  may  then  be  ex- 
pressed as  so  many  ampere-turns.  The  ampere-turn  is  the  unit 
of  magnetomotive  force  used  in  practice;  the  relation  between 
the  gilbert  and  ampere-turn  is 

1  gilbert  =0.79578  ampere-turn. 

114.  Magnetic  Reluctance.  — The  analogy  between  the  rela- 
tion connecting  the  drop  of  magnetic  potential  and  magnetomotive 
force  and  the  relation  between  resistance  drop  and  electromotive 
force  may  be  extended  still  further.  Consider  first  the  simple  case 
of  a  coil  of  insulated  wire  uniformly  wound  around  a  closed 
anchor  ring,  such  as  described  in  Article  110.  Let  the  cross  sec- 
tion of  this  ring  be  A,  and  the  length  of  the  mean  circumference 
of  the  ring  be  /.  From  symmetry,  the  lines  of  force  produced  by 
an  electric  current  in  the  coil  wound  on  this  ring  will  be  circles 
concentric  with  the  center  of  the  ring,  and  each  line  of  force  must 
therefore  link  all  the  turns  of  the  coil.  The  field  intensity  will 
also  have  the  same  value  for  every  point  on  any  one  of  these 
circles.  Hence,  calling  H  the  field  intensity  at  any  point  on  the 


208  ELECTRICAL  ENGINEERING 

mean  circumference  of  the  ring,  the  drop  of  magnetic  potential 
around  this  circumference  is 

When  the  radius  of  the  ring  is  large  compared  with  the  radius  of 
the  section  A,  the  field  intensity  at  the  mean  circumference 
may  be  taken  as  the  average  field  intensity  over  this  area  A. 
Hence,  calling  p  the  permeability  of  the  iron  forming  the  ring, 
the  average  flux  density  over  the  area  A  is  ^H,  and  therefore 
the  total  flux  or  number  of  lines  of  induction  through  A  is  <f>  =p.  HA. 
Hence  A77 

t*£?™ 

_l_  (9) 

The  expression  -  -  is  analogous  to  the  expression  for  the  re- 
sistance of  a  wire  of  length  I,  cross  section  A  and  specific  resist- 
ance — ,  and  it  is  therefore  called  the  reluctance  of  the  magnetic 

circuit  formed  by  the  ring. 

The  reluctance  of  any  portion  of  a  magnetic  circuit  has  no 
meaning  unless  the  two  ends  of  the  given  portion  of  the  circuit 
where  the  lines  of  induction  enter  and  leave  it  are  magnetic 
equipotential  surfaces  and  the  total  number  of  lines  of  induction 
through  each  cross  section  of  the  given  portion  of  the  circuit  is 
the  same.  When  these  conditions  are  satisfied,  the  reluctance 
R  may  be  defined  as  the  ratio  of  the  difference  of  magnetic 
potential  Vm  between  the  two  end  faces  of  the  given  portion  of 
the  circuit  to  the  total  flux  of  induction  <£  "through  the  circuit,  i.e., 

R=^  (10) 

9 
When  the  difference  of  magnetic  potential  is  expressed  in  gilberts 

and  the  flux  in  maxwells,  the  unit  of  reluctance  is  called  the 
oersted.  The  relation  expressed  by  equation  (10)  is  of  exactly 
the  same  form  as  Ohm's  Law  for  an  electric  circuit ;  it  is  therefore 
sometimes  called  "  Ohm's  Law  for  a  magnetic  circuit."  Magnetic 
flux,  magnetomotive  force  and  magnetic  reluctance  are  strictly 
analogous  to  electric  current,  electromotive  force  and  electric 
resistance  respectively,  except  that  no  energy  is  required  to  maintain 
a  magnetic  flux  through  a  reluctance,  while  energy  is  always  required 
to  maintain  an  electric  current  through  a  resistance. 


ELECTROMAGNETISM  209 

Since  lines  of  magnetic  induction  are  always  closed  loops, 
the  flux  of  magnetic  induction  coming  up  to  any  surface  must 
always  equal  the  flux  of  induction  leaving  that  surface.  There- 
fore at  any  junction  in  a  network  of  magnetic  circuits 

2<£=o  (11) 

This  is  the  same  as  Kirchhoff's  first  law  for  an  electric  circuit. 
Similarly,  since  the  drop  of  magnetic  potential  Vm  around  any 
closed  loop  is  equal  to  the  total  magnetomotive  force  in  this 
path, 

2<f>R=24irNI  (Ha) 

where  R  is  the  reluctance  of  any  closed  tube  of  induction  and  <j> 
is  the  flux  through  this  tube.  This  last  relation  is  the  same  as 
Kirchhoff's  second  law  for  a  network  of  electric  circuits. 

The  difficulty  in  applying  these  laws  to  a  magnetic  circuit 
arises  from  the  fact  that  the  magnetic  flux  is  not  confined  to 
approximately  geometrical  lines  like  the  currents  in  a  network 
of  insulated  wires,  but  in  general  fills  all  space  surrounding  the 
coils  which  establish  the  magnetomotive  forces ;  also,  when  there 
is  iron  in  the  circuit  the  permeability  depends  on  the  flux  density 
and  the  previous  history  of  the  iron.  (The  distribution  of  magnetic 
flux  in  and  around  an  iron  circuit  is  analogous  to  the  distribution 
of  current  in  and  around  an  uninsulated  mass  of  copper  of  the 
same  shape  as  the  iron  circuit  immersed  in  a  liquid  having  a 
conductivity  about  equal  to  that  of  carbon.)  Ohm's  Law  for  the 
magnetic  circuit,  however,  tells  us  that  in  order  to  obtain  a  large 
flux  with  the  least  number  of  ampere-turns  it  is  necessary  to 
provide  a  path  of  low  reluctance  for  the  lines  of  induction.  Hence 
in  nearly  all  electric  machinery  a  closed  or  nearly  closed  iron  or 
steel  circuit  is  provided  for  the  lines  of  induction,  since  iron  and 
steel  have  a  high  permeability,  and  therefore  for  the  same  dimen- 
sions a  much  less  reluctance  than  a  non-magnetic  substance. 
Only  in  the  special  case  of  the  uniformly  wound  circular  ring 
discussed  above,  however,  are  the  lines  of  induction  confined 
entirely  to  an  iron  circuit ;  in  general  a  certain  number  also  exist 
in  the  air  and  in  whatever  other  substances  are  in  the  vicinity 
of  the  iron  circuit.  For  example,  in  the  case  of  a  dynamo,  a 
certain  percentage  of  the  total  number  of  lines  of  induction  estab- 
lished by  the  field  coils  "  leak  "  around  through  the  air  from 
one  pole  to  the  next  without  linking  the  armature  conductors. 
The  predetermination  of  the  ratio  of  the  total  flux  to  the  useful 


210  ELECTRICAL  ENGINEERING 

flux,  i.e.,  the  ratio  of  the  total  number  of  lines  of  induction  to  the 
number  which  link  the  armature  conductors,  can  be  made  only 
very  roughly;  this  ratio,  which  is  called  the  "leakage  factor," 
may,  however,  be  determined  by  experiment ;  in  modern  dynamos 
is  found  to  vary  from  1.1  to  1.5,  depending  upon  the  arrangement 
of  the  field  magnets. 

115.  Calculation  of  Ampere-Turns  Required  to  Establish  a 
Given  Flux.  —  The  following  example  will  illustrate  the  way  in 
which  the  number  of  ampere-turns  required  to  establish  a  given 
flux  may  be  calculated  to  a  rough  degree  of  approximation. 
Let  it  be  required  to  find  the  number  of  ampere-turns  necessary 
to  establish  a  total  flux  <j>  through  the  armature  of  a  two-pole 
dynamo.  Let  the  armature  be  made  of  sheet  steel  punchings 
and  the  field  cores,  pole  shoes,  and  field  yoke  be  a  single  piece  of 
cast  iron. 
Let 

Ag  =area  of  the  air  gap  under  each  pole. 
Aa  =the  mean  cross  section  of  the  path  of  the  lines  of  induc- 
tion through  the  armature. 

Af=the  mean  cross  section  of  the  path  of  the  lines  of  in- 
duction through  the  field. 
lg   =the  radial  depth  of  the  air  gap ;  usually  called  the  length 

of  the  air  gap. 
la   =the  mean  length  of  the  path  of  the  lines  of  induction 

through  the  armature. 
If    =the  mean  length  of  the  path  of  the  lines  of  induction 

through  the  field. 
k    =the  leakage  factor. 

Then  the  flux  density  in  the  air  gap  is   -?—  and  this  is  also  equal 

Ag 

to  the  field  intensity  in  the  air  gap,  that  is  Hg=-^- .     Hence  the 

Ag 

rt        I        7 

drop  of  magnetic  potential  across  both  air  gaps  is  2  Hg  lg  =  — ?_J! . 

,  Af 

The  flux  density  in  the  armature  is  -?- ;   from  the  B  —  H  curve 

Am 

for  sheet  steel  punchings  find  the  corresponding  value  of  H; 
call  this  value  Ha.  Then  the  drop  of  magnetic  potential  through 
the  armature  is  Ha  la.  The  total  flux  through  the  field  will  be 


ELECTROMAGNETISM  211 

k  <h,  and  therefore  the  flux  density  in  the  field  will  be  -2;  from 

At 

the  B—  H  curve  for  cast  iron  find  the  corresponding  value  of  H; 

call  this  value  Hf.  Then  the  drop  of  magnetic  potential  through 
the  field  will  be  Hf  lf.  Hence,  equating  the  total  drop  of  magnetic 
potential  around  the  entire  magnetic  circuit  to  the  total  magneto- 
motive force  ^irNI  linked  by  the  mean  path  of  the  flux,  we  have 

Ha  la  +  Hf  lf  =4  TT  NI  (12) 


9 
all  in  c.  g.  s.  electromagnetic  units.     From  this  relation  the  number 

of  ampere-turns  can  be  calculated.  For  calculations  of  this  sort 
it  is  much  more  convenient  to  have  the  B—  H  curves  plotted 
in  c.  g.  s.  lines  per  square  inch  as  ordinates  against  field  intensity 
in  ampere-turns  per  inch.  When  (j>  is  expressed  in  maxwells, 
H  is  expressed  in  ampere-turns  per  inch,  I  in  inches,  A  in  square 
inches  and  /  in  amperes,  equation  (8)  becomes 


It  should  be  noted  that  the  ampere-turns  thus  calculated  are  the 
net  ampere-turns  required  to  establish  the  given  flux  through 
the  armature,  that  is,  the  difference  between  the  ampere-turns 
which  must  be  on  the  field  coils  and  the  back  ampere-turns  of 
the  armature.  (See  under  Armature  Reaction  in  any  text-book 
on  dynamo-electric  machinery.) 

W116.  Self  and  Mutual  Induction.  —  When  the  current  in  any 
electric  circuit  varies  with  time,  the  magnetic  field  produced  by 
this  current  also  varies  with  time;  hence  any  electric  circuit 
which  is  linked  by  the  lines  of  induction  due  this  varying  current 
will  have  an  electromotive  force  induced  in  it»  In  particular,  the 
circuit  in  which  the  current  in  question  is  flowing  will  have  an 
electromotive  force  induced  in  it  ;  the  electromotive  force  induced 
in  a  circuit  due  to  the  variation  of  the  current  in  this  circuit  is 
called  the  electromotive  force  of  self  induction  in  this  circuit- 
Similarly,  the  electromotive  force  induced  in  any  other  circuit  2 
as  the  result  of  the  variation  of  the  current  in  any  circuit  1  js 
called  the  electromotive  force  of  mutual  induction  in  circuit  1 
due  to  circuit  2.  If  a  varying  current  in  any  circuit  1  produces 
an  electromotive  force  in  some  other  circuit  2,  then  a  varying 
current  in  2  will  likewise  produce  an  electromotive  force  in  1  ; 
hence  the  name  "  mutual  "  for  such  electromotive  forces. 


212  ELECTRICAL  ENGINEERING 

The  numerical  value  of  the  ratio  of  the  electromotive  force 
induced  in  a  circuit,  due  to  the  change  of  the  current  in  this 
circuit,  to  the  time  rate  of  this  change,  is  called  the  coefficient  of 
self  induction  or  the  self  inductance  of  this  circuit.  That  is,  when 

di 

the  current  in  the  given  circuit  varies  at  the  rate  —  and,  as   a 

dt 

result  of  this  variation,  an  electromotive  force  e  is  induced  in 
this  circuit,  then  the  self  inductance  of  this  circuit  is 

L=— 

<&_  (13) 

~dt 

When  the  electromotive  force  is  expressed  in  abvolts,  the  current 
in  abamperes  and  time  in  seconds,  the  unit  of  self  inductance  is 
called  the  abhenry;  when  these  quantities  are  expressed  in  volts, 
amperes  and  seconds  respectively,  the  unit  of  self  inductance  is 
called  the  henry.  A  millihenry  is  the  one-thousandth  part  of  a 
henry.  These  units  are  therefore  related  as  follows  : 

1  henry  =109  abhenries 

1  millihenry        =106  abhenries 

Similarly,  the  numerical  value  of  the  ratio  of  the  electro- 
motive force  elz  induced  in  any  circuit  1,  due  to  a  change  in  the 
current  i2  in  any  other  circuit  2,  to  the  time  rate  of  the  change 
of  this  current  iz,  is  called  the  coefficient  of  mutual  induction  or  the 
mutual  inductance  M12  of  circuit  2.  with  respect  to  circuit  1 ;  that  is 

^12=— 

di,  (14) 

dt 

Mutual  inductance  is  expressed  in  the  same  unit  as  self  inductance* 
The  self  inductance  of  a  circuit  may  also  be  expressed  in  terms 
of  the  number  of  linkages  (see  Article  105)  between  the  circuit 
and  the  flux  produced  by  the  current  in  it.  From  the  funda- 
mental law  of  electromagnetic  induction,  a  change  in  the  number 
of  linkages  between  a  circuit  and  the  flux  linking  it  induces  in 
this  circuit  an  electromotive  force  equal  to  the  time  rate  of  change 
of  these  linkages.  Hence,  calling  d\  the  change  in  the  number 
of  linkages  between  the  circuit  and  the  flux  due  to  a  change  in 
the  current  in  the  circuit  by  an  amount  di,  we  have  as  another 
expression  for  the  electromotive  force  of  self  induction 


ELECTROMAGNETISM  213 

d\ 

e= — 
dt 

Equating  this  value  of  the  self-induced  electromotive  force  to 
that  given  by  equation  (13)  we  have  that  the  self  inductance  in 
abhenries  is 

L-%  (15) 

di 

where  X  is  expressed  in  maxwells  and  i  in  abamperes.  Hence  the 
self  inductance  of  a  circuit  is  equal  to  the  change  of  the  linkages 
between  the  circuit  and  the  flux  threading  it  per  unit  change 
in  the  current  in  this  circuit.  As  will  be  proved  presently,  when 
the  permeability  of  every  body  in  the  magnetic  field  is  constant 
and  the  circuit  remains  unaltered  in  size  and  shape,  the  linkages 
between  the  circuit  and  the  flux  threading  it  due  to  the  current 
in  this  circuit  are  directly  proportional  to  the  strength  of  the  current 
in  the  circuit;  under  these  conditions,  therefore,  the  self  induc- 
tance is  a  constant  of  the  circuit  (for  a  given  distribution  of  current) 
equal  to  the  number  of  linkages  between  this  circuit  and  the  flux 
produced  by  unit  current  in  it. 

Similarly,  the  mutual  inductance  in  abhenries  of  a  circuit  2 
with  respect  to  any  other  circuit  1  is  equal  to  the  change  of  the 
linkages  X12  of  circuit  1  by  the  flux  due  to  the  current  i2  in  2  per 
unit  change  in  the  current  in  2,  i.e., 

M      d\,  (16) 

JK*  12  —  . 

ai3 

where  X12  is  in  maxwells  and  ia  in  abamperes.  When  the  per- 
meability of  every  body  in  the  magnetic  field  is  constant  and  the 
two  circuits  are  fixed  with  respect  to  each  other  and  remain  un- 
altered in  size  and  shape,  the  mutual  inductance  of  one  circuit 
with  respect  to  another  is  constant  (for  a  given  distribution  of  the 
currents)  and  equal  to  the  number  of  linkages  between  one  circuit  and 
the  flux  produced  by  unit  circuit  in  the  other;  moreover,  as  will  be 
proved  presently,  the  mutual  inductance  of  one  circuit  with  respect 
to  the  other  is  equal  to  the  mutual  inductance  of  the  second  circuit 
with  respect  to  the  first. 

117.  Proof  of  the  Relation  between  Inductance  and  Linkages.— 
The  intensity  at  any  point  of  the  magnetic  field  due  to  an  electric 

current  is  given  by  the  equation  (see  Article  71)  H  =i  |       vl      ' 


214  ELECTRICAL  ENGINEERING 

where  i  is  the  strength  of  the  current  in  the  circuit,  dl  any 
elementary  length  of  the  circuit,  r  the  distance  of  the  point  in 
question  from  the  elementary  length  dl  and  0  the  angle  between 
the  line  drawn  from  the  point  to  dl  and  the  direction  of  dl,  and  the 

symbol  J  represents  the  vector  integral  of  the  expression 
—  around  the  entire  circuit.  As  long  as  the  shape  and  posi- 
tion of  the  circuit  remain  unaltered,  the  quantity  under  the  integral 
sign  remains  unaltered ;  hence  the  field  intensity  .at  any  point  due 
directly  to  a  current  in  a  given  circuit  is  proportional  to  the 
strength  of  this  current,  no  matter  what  the  shape  or  size  of  the 
circuit  may  be,  provided  the  shape  and  size  remain  unaltered. 
The  field  intensity  at  any  point  in  the  surrounding  region  due  to 
the  magnetic  poles  which  may  be  induced  by  this  current  on  any 
magnetic  bodies  in  the  vicinity  will  also  be  proportional  to  the 
strength  of  this  current,  provided  the  permeabilities  of  these  bodies 
are  constant.  Under  these  conditions  the  directions  of  the  result- 
ant lines  of  force  and  the  resultant  lines  of  induction  established 
by  the  electric  current  will  remain  unaltered  when  the  strength 
of  the  current  is  changed,  but  their  number  crossing  any  surface 
will  vary  directly  as  the  strength  of  the  current.  The  lines  of 
force  and  the  lines  of  induction  will  also  coincide  in  direction. 
Hence  the  number  of  lines  of  induction  crossing  any  area  in  the 
magnetic  field  due  to  a  current  in  a  single  circuit  is  directly  pro- 
portional to  the  strength  of  the  current  in  this  circuit,  provided 
all  the  surrounding  bodies  have  a  constant  permeability. 

In  particular,  the  number  of  lines  of  induction  threading  the 
circuit  itself  is  proportional  to  the  strength  of  the  current  in  the 
circuit.  Therefore,  calling  i  the  strength  of  the  current  in  the 
circuit  at  any  instant,  the  number  of  lines  of  induction  threading 
the  circuit  at  this  instant  is  proportional  to  i,  and  therefore  the 
number  of  linkages  X  between  the  circuit  and  the  flux  due  to  the 
current  in  it  is  proportional  to  i,  that  is 

\=Ai 

where  A  is  a  constant  depending  upon  the  shape  and  size  of  the 
circuit  and  the  magnetic  nature  of  the  bodies  in  the  field,  but  in- 
dependent of  the  strength  of  the  current,  provided  the  permeability 
of  every  body  in  the  field  is  constant.  Note  that  this  constant  A 
is  equal  to  the  number  of  linkages  between  this  circuit  and  the 


ELECTROMAGNETISM  215 

flux  produced  by  unit  current  in  it,  for  when  i  =  l,  X=A.  More- 
over, from  equation  (15),  this  constant  is  the  self  inductance  of 

the  circuit,  for  —  =A.     The  self  inductance  of  a  circuit  may  be 
di 

looked  upon  as  a  constant  which  represents  the  extent  and  dis- 
tribution of  the  magnetic  field  due  a  current  in  the  circuit,  but 
is  independent  of  the  value  of  the  field  intensity  provided  the 
permeabilities  of  all  bodies  in  the  field  are  constant. 

Similarly,  the  number  of  lines  of  induction  threading  a  given 
circuit  1  due  to  a  current  ia  in  another  circuit  2  is  proportional  to 
the  current  in  2,  and  therefore  the  number  of  linkages  X12  of 
the  circuit  1  by  the  flux  due  to  the  current  in  2  is  proportional  to 

the  current  i2,  that  is 

A,  2  =Bi2 

where  B  is  a  constant  depending  upon  the  size  and  shape  of  the 
two  circuits,  their  relative  positions  and  the  nature  of  the  bodies 
in  the  magnetic  field,  but  independent  of  the  strength  of  the 
current,  provided  the  permeability  of  every  body  in  the  magnetic 
field  is  constant.  Note  that  this  constant  B  is  equal  to  the  number 
of  linkages  of  the  circuit  1  by  the  flux  due  to  unit  current  in 
2,  for  when  i'2  =  l,  X12=J5.  Moreover,  from  equation  (16),  this 
constant  is  the  mutual  inductance  of  circuit  2  with  respect  to 

circuit  1 ,  for  — -  =  B.     In  the  case  of  two  circuits  three  con- 
di2 

stants  are  required  to  represent  the  extent  and  distribution  of 
the  magnetic  field  due  to  the  currents  in  these  circuits,  the  self 
inductance  of  each  circuit  and  a  single  coefficient  of  mutual  in- 
ductance (see  Article  120). 

118.  Magnetic  Energy  of  an  Electric  Current. —  Since  when  the 
strength  of  the  current  in  a  circuit  is  increased  a  back  electromotive 
force  is  induced  in  the  circuit  (due  to  the  increase  in  the  flux 
linking  the  circuit),  a  certain  amount  of  energy  is  required  to 
establish  an  electric  current,  just  as  energy  is  required  to  establish 
a  water  current,  i.e.,  to  set  a  mass  of  water  in  motion.  This 
energy  comes  from  the  source  of  the  electromotive  force  which 
establishes  the  current,  just  as  the  energy  for  accelerating  the 
velocity  of  a  body  comes  from  the  source  of  the  mechanical  force 
which  sets  the  body  in  motion.  Since  the  magnetic  field  sur- 
rounding the  circuit  may  be  considered  as  the  source  of  the  back 
electromotive  force  in  the  circuit,  the  energy  required  to  establish 


216  ELECTRICAL  ENGINEERING 

the  electric  current  may  be  said  to  be  "  stored  "  in  the  magnetic 
field,  just  as  the  kinetic  energy  of  a  moving  body  may  be  looked 
upon  as  stored  in  the  body  itself  as  a  consequence  of  its  inertia, 
which  may  be  looked  upon  as  the  source  of  the  opposition  which 
the  body  offers  to  being  set  in  motion.  Again,  when  the  strength 
of  the  current  in  a  circuit  decreases,  an  electromotive  force  is 
induced  in  the  circuit  in  the  same  direction  as  the  current,  and 
therefore  a  certain  amount  of  the  energy  stored  in  the  magnetic 
field  of  the  current  is  converted  into  some  other  form  of  energy 
by  means  of  the  current  in  the  circuit  (e.g.,  heat  energy),  just  as 
when  the  velocity  of  a  moving  body  decreases  a  definite  amount 
of  its  kinetic  energy  is  converted  into  other  forms  of  energy  (e.g., 
heat  energy). 

In  general,  no  matter  how  a  magnetic  field  may  be  formed, 
whether  by  means  of  electric  currents  or  magnetic  poles,  a 
definite  amount  of  energy  is  required  to  establish  it  and  when 
the  field  is  destroyed  this  energy  appears  in  some  other  form. 
Every  magnetic  field  therefore  represents  a  certain  amount  of 
"  stored,"  or  potential,  energy.  This  energy  of  a  magnetic  field 
is  called  magnetic  energy;  when  the  magnetic  field  is  due  to  electric 
currents  this  energy  is  also  called  the  electrokinetic  energy  of  the 
currents,  the  latter  name  being  due  to  the  analogy  between  this 
energy  and  the  kinetic  energy  of  a  moving  body. 

The  magnetic  or  electrokinetic  energy  of  the  magnetic  field 
set  up  by  a  current  in  a  circuit  may  be  expressed  in  terms  of  the 
current  strength  and  the  inductance  of  the  circuit.  Let  i  be  the 
current  in  abamperes  in  the  circuit  at  any  instant  and  let  L 
be  the  self  inductance  of  the  circuit,  in  abhenries  ;  then  the 
linkages  X  between  the  circuit  and  the  flux  due  to  the  cur- 
rent in  it  at  this  instant  are  X  =  Li.  The  back  electromotive 

force  due  to  an  increase  di  in  the  current  in  time  dt  is  then  e  =L  —  , 

dt 

provided  L  is  constant,  and  therefore  the  energy  transferred  to 
the  magnetic  field  by  the  current  in  time  dt  is  * 

dW=eidt  =  Lidi 

Therefore  the  total  amount  of  energy  stored  in  the  magnetic  field 
of  the  current  when  the  strength  of  the  current  increases  from  0 
to  any  value  i  is 

TF  =  /     Lidi=%Li*  (17) 

J 


o 


ELECTROMAGNETISM  217 

The  energy  stored  in  the  field  when  the  current  is  established 
is  equal  to  the  energy  returned  to  the  circuit  when  the  current  is 

interrupted,  for 

/o  ri 

Lidi  —  ~  I   Lidi 
i  Jo 

provided  L  is  constant.  The  energy  returned  to  the  circuit  from 
the  magnetic  field  gives  rise  to  an  electromotive  force  which 
tends  to  keep  the  current  flowing  in  its  original  direction,  just  as 
when  the  motion  of  a  body  is  opposed  in  any  way,  the  kinetic 
energy  of  the  body  tends  to  keep  it  moving  in  its  original  direction. 

The  instantaneous  value  of  this  electromotive  force  at  the  in- 
stant a  circuit  is  opened  is  generally  sufficiently  great  to  establish 
a  momentary  current  through  the  air  or  whatever  else  separates 
the  ends  of  the  opened  circuit,  and  consequently  the  current 
continues  to  flow  for  a  fraction  of  a  second  across  the  space  be- 
tween the  open  ends  of  the  circuit,  producing  the  familiar  spark 
or  arc  which  occurs  when  a  circuit  is  opened  (unless  certain  special 
precautions  are  taken).  The  energy  which  produces  this  spark 
is  the  energy  which  is  stored  in  the  magnetic  field  around  the  cir- 
cuit when  the  current  is  established,  and  is  generally  not  sufficient 
to  maintain  for  more  than  a  fraction  of  a  second  a  current  through 
the  high  resistance  of  the  insulator  between  the  open  ends  of  the 
circuit.  However,  whenever  a  current  is  established  through  an 
insulator  its  resistance  falls  to  a  comparatively  low  value,  at  which 
it  remains  as  long  as  the  current  through  it  is  maintained.  Hence, 
when  the  gap  formed  by  opening  the  circuit  is  short,  the  electro- 
motive force  originally  in  the  circuit  may  be  sufficient  to  maintain 
a  comparatively  large  current  through  the  gap.  It  is  this  heavy 
current  established  across  the  gap  by  the  electromotive  force 
originally  in  the  circuit  which  constitutes  the  so-called  arc,  and 
which  rapidly  burns  away  the  conductors  at  the  gap  unless  a 
suitable  switch  or  circuit-breaker  is  provided  for  opening  the 
circuit.  When  the  current  in  the  circuit  is  large  or  the  electro- 
motive force  in  the  circuit  high,  the  circuit-breaker  must  open  the 
circuit  rapidly  and  make  a  long  gap  between  the  opened  ends. 

It  should  be  noted  that  the  above  formula  for  the  energy 
required  to  establish  a  magnetic  field  is  deduced  on  the  assump- 
tion that  the  self  inductance  is  constant,  which  is  true  only  when 
the  permeability  of  every  body  in  the  magnetic  field  is  constant. 
In  the  case  of  iron  and  other  magnetic  bodies  the  permeability 


218  ELECTRICAL  ENGINEERING 

is  not  constant,  and  in  addition  to  the  transfer  of  energy  to  and 
from  the  magnetic  field,  a  certain  amount  of  energy  is  converted 
into  heat  energy  (due  to  hysteresis,  see  Article  126),  both  when 
the  current  is  established  and  also  when  the  current  is  inter- 
rupted, or  whenever  there  is  any  change  whatever  in  the  current 
strength. 

119.  Analogy  between  the  Magnetic  Energy  of  an  Electric 
Current  and  the  Kinetic  Energy  of  a  Moving  Column  of  Water.— 
As  already  noted,  the  magnetic  energy  of  an  electric  current  is  in 
many  ways  analogous  to  the  kinetic  energy  of  a  moving  column 
of  any  liquid,  as  for  example,  a  current  of  water  in  a  pipe.  The 
kinetic  energy  of  such  a  "  water  current  "  is  the  work  required 
to  set  the  water  in  motion ;  similarly,  the  magnetic  energy  of  an 
electric  current  may  be  looked  upon  as  the  work  required  to  "  set 
the  electricity  in  the  circuit  in  motion."  When  the  water  comes 
to  rest,  its  kinetic  energy  is  converted  into  some  other  form, 
principally  heat  energy.  Similarly,  when  an  electric  current  is 
interrupted,  thus  causing  the  "  electricity  to  come  to  rest,"  the 
magnetic  energy  of  the  current  is  converted  into  some  other  form, 
e.g.,  heat  energy  in  the  conductors  and  in  the  spark  which  appears 
at  the  break  in  the  circuit. 

The  kinetic  energy  of  a  moving  column  of  water  can  also  be 
expressed  by  a  formula  similar  to  the  formula  W  =  ^  Li?  for  the 
magnetic  energy  of  an  electric  current.  Consider  a  stream  of 
water  flowing  through  a  pipe  of  uniform  cross  section  of  A  sq. 
cm.  and  let  the  water  completely  fill  the  pipe;  let  /  be  the  length 
in  cm.  of  a  given  section  of  the  pipe,  and  let  V  be  the  linear  velocity 
of  the  water  in  cm.  per  second.  Then,  since  the  mass  of  1  cu. 
cm.  of  water  is  1  gram,  the  kinetic  energy  of  the  water  in  the  given 
length  of  pipe  is  i  (I A)  V2.  Let  J  be  the  volume  or  "  quantity  " 
of  water  that  flows  across  any  section  of  the  pipe  in  one  second, 
then  J  =  AV.  Hence  the  kinetic  energy  of  the  water  column  may 

also  be  expressed  as  \  —  J2  =  \  KJ2,  where   K  =  -  is  a  constant 

-TL  -A. 

depending  upon  the  dimensions  of  the  pipe.  The  analogy  between 
this  last  expression  and  the  formula  £  Li2  for  the  magnetic  energy 
of  an  electric  current  is  at  once  evident,  when  an  electric  current 
is  considered  as  the  quantity  of  electricity  per  second  flowing  across 
any  section  of  the  wire.  Note,  however,  that  while  the  factor  K 
in  the  expression  for  the  kinetic  energy  of  the  water  current 


ELECTROMAGNETISM  219 

depends  only  upon  the  length  and  cross  section  of  the  given  length 
of  pipe,  the  factor  L  in  the  expression  for  the  magnetic  or  elec- 
trokinetic  energy  of  an  electric  current  depends  not  only  upon  the 
length  and  cross  section  but  also  upon  the  shape  of  the  wire  form- 
ing this  circuit,  upon  the  shape  of  the  circuit,  and  upon  the  nature 
of  the  surrounding  bodies. 

120.  Magnetic  Energy  of  Two  or  More  Electric  Currents.— As 
already  noted  (Article  116),  when  the  strength  of  the  current  in  an 
electric  circuit  changes  the  corresponding  change  in  its  magnetic 
field  induces  an  electromotive  force  not  only  in  this  circuit  but 
also  in  every  other  circuit  in  the  vicinity.  Consequently  if  a 
current  already  exists  in  any  neighboring  circuit,  or  if  the  induced 
electromotive  force  establishes  a  current  in  such  a  circuit,  there 
will  be  a  transfer  of  energy  from  one  circuit  to  the  other.  Whether 
work  is  done  on  the  current  or  is  done  by  the  current  in  a  given 
circuit  depends  upon  the  relative  direction  of  the  current  and  the 
electromotive  force  induced  in  this  circuit.  In  any  problem 
dealing  with  the  mutual  effects  of  -two  or  more  circuits  it  is  there- 
fore necessary  to  adopt  some  convention  in  regard  to  the  algebraic 
signs  of  the  various  currents ;  this  is  conveniently  done  by  choosing 
a  given  sense  of  the  lines  of  induction  as  positive  (e.g.,  left  to  right) 
and  to  consider  the  current  in  any  circuit  as  positive  if  the  lines 
of  induction  which  it  sets  up  thread  this  circuit  in  this  same  sense, 
negative  if  these  lines  thread  the  circuit  in  the  opposite  sense. 
For  example,  when  two  coils  of  wire  are  placed  side  by  side,  the 
currents  in  the  two  coils  are  to  be  considered -in  the  same  sense 
if  the  lines  of  induction  set  up  by  the  current  in  one  coil  thread 
the  other  in  the  same  direction  as  the  lines  of  induction  set  up  by 
the  current  in  the  latter. 

Consider  first  the  simple  case  of  two  circuits  in  the  vicinity 
of  each  other,  and  let  the  circuits  be  fixed  in  size,  shape  and  rel- 
ative position,  and  let  the  permeability  of  every  body  in  the  field 
be  constant.  Let  I/x  and  L2  be  the  self  inductances  of  the  two 
circuits  and  M12  the  mutual  inductance  of  circuit  2  with  respect 
to  circuit  1  and  M21  the  mutual  inductance  of  1  with  respect  to  2 ; 
let  iv  and  i2  be  the  currents  in  the  two  circuits  at  any  instant. 
Then,  from  Article  117,  the  total  linkages  of  the  two  circuits  by 
the  flux  threading  them  are  respectively 

i2  . 


220  ELECTRICAL  ENGINEERING 

where  the  X's  and  i's  may  be  positive  or  negative.  Let  the  two 
currents  increase  by  the  amounts  dit  and  di2  in  time  dt;  then  the 
back  electromotive  forces  induced  in  the  two  circuits  are  re- 
spectively 


dt  dt  dt 

d\2     T  di2  dit 

e2=  -  =L/2  --  \-M2l  — 
dt          dt  dt 

and  the  amounts  of  energy  stored  in  the  magnetic  field  by  the 
respective  currents  are 

dWl  =elildt=Llildil  +  Ml2i^di2 

d  W2  =  e2i2dt  =  L2i2di2  +  M^izdi^ 

The  total  amount  of  energy  stored  in  the  magnetic  field  by 
two  currents  when  they  increase  from  zero  to  their  final  values 
/!  and  72  may  be  calculated  as  follows.  First,  let  circuit  2  be  open, 
so  that  no  current  can  flow  in  it  ;  under  these  conditions  i2  =0 
and  di2  =0,  and  therefore  the  work  done  by  the  current  in  circuit 
1  when  the  current  in  this  circuit  increases  from  zero  to  1^  is 


o 

Now  keep  the  current  in  circuit  1  constant  and  let  the  current  in 
2  increase  from  0  to  72 ;  under  these  conditions  i\  =Il  and  di^  =0, 
and  therefore  the  work  done  by  the  current  in  circuit  1  is 


Ml2Ildi2=MlJ1I2 
and  the  work  done  by  the  current  in  circuit  2  is 


Hence  the  total  work  done  by  the  two  currents  in  establishing 
the  magnetic  field  corresponding  to  the  final  values  of  the  currents 
/!  and  72  is 

Note  that  this  formula  does  not  contain  the  coefficient  M2l. 
The  explanation  of  this  is  the  fact,  already  noted  several  times, 
that  the  mutual  inductance  of  one  circuit  with  respect  to  another 
is  the  same  as  the  mutual  inductance  of  the  second  circuit  with 
respect  to  the  first  circuit.  To  prove  this,  let  the  current  in  2  now 
be  kept  constant  and  let  the  current  in  1  be  decreased  to  zero; 


ELECTROMAGNETISM  221 

under  these  conditions  i'2  =72  and  di2  =0  ;  and  therefore  the  work 
done  on  the  current  in  1  by  the  magnetic  field  is 


and  the  work  done  on  the  current  in  2  by  the  magnetic  field  is 

21/A  =M2JJ2 


Now  open  circuit  1  and  let  the  current  in  2  fall  to  zero  ;  under  these 
conditions  i\  =0  and  di^  =0,  and  therefore  the  work  done  on  the 
current  in  2  by  the  magnetic  field  when  this  current  falls  to  zero  is 

-  f°  L2i2di2=±L2P2 

J  /2 
Hence  the  total  work  done  by  the  magnetic  field  when  it  disappears 

is 

W  =%LJ\  +  M2JJ2+iL2I22  (206) 

From  the  Principle  of  the  Conservation  of  Energy,  the  energy 
stored  in  the  field  when  it  is  established  must  be  equal  to  the  work 
done  by  the  field  when  it  disappears,  hence  the  expression  for  the 
energy  stored  in  the  field  must  be  equal  to  the  expression  for  the 
energy  given  back  by  the  field  ;  therefore 

M21=M12 

It  also  follows  from  the  Principle  of  the  Conservation  of  Energy 
that  the  total  amount  of  energy  stored  in  the  magnetic  field  by 
the  two  currents  is  independent  of  the  manner  in  which  these  two 
currents  are  established,  and  therefore  the  expression 

W  =iZ^7  +  MI&  +  iLA  (21) 

is  a  perfectly  general  one  for  the  energy  of  the  magnetic  field  due 
to  the  currents  t\  and  i2  in  two  circuits  which  have  constant  self 
inductances   Lv  and   L2  and  a  constant  mutual  inductance  M. 
Equation  (21)  may  also  be  written 

W=  i  (Z,k  +  M12t2)  i,  +  \  (L2i2+  MM  i2 
which,  in  turn,  from  equation  (18),  may  be  written 

w  =4x^+4x^2 

By  exactly  similar  reasoning  it  can  be  shown  that  the  energy 
of  the  magnetic  field  due  to  the  electric  currents  in  any  number 
of  circuits  is 

TT  =  iSXt  (22) 

where  i  is  the  current  in  any  circuit  and  X  is  the  number  of  link- 
ages between  this  circuit  and  the  total  flux  which  threads  it,  and 
the  summation  includes  all  the  circuits  in  the  field. 


222  ELECTRICAL   ENGINEERING 

It  should  be  noted  that  all  the  formulas  in  this  article  are 
based  upon  the  assumption  that  every  body  in  the  magnetic  field 
has  a  constant  permeability. 

121.  Calculation  of  Inductance.  —  Skin  Effect — When  the 
permeability  of  every  body  in  the  magnetic  field  is  constant,  it  is 
in  general  possible  to  obtain  an  expression  for  the  inductance  of  a 
circuit  in  terms  of  the  dimensions  of  the  circuit.  To  obtain  an 
exact  expression  for  this  quantity,  however,  it  is  necessary  to 
consider  the  conductors  forming  the  circuit  as  divided  into  current 
filaments  (see  Article  101)  and  to  determine  the  back  electromotive 
force  induced  in  each  of  these  filaments  when  the  current  changes. 
The  inductance  of  a  circuit  therefore  depends  upon  the  distribu- 
tion of  the  current  in  the  conductors,  and,  as  already  noted 
(Article  101),  the  distribution  of  the  current  in  a  conductor  depends 
upon  the  rapidity  with  which  the  current  varies  with  time.  The 
explanation  of  this  will  now  be  apparent ;  when  the  conductor 
has  a  large  cross  section  or  is  a  magnetic  substance  (e.g.,  a  steel 
rail)  the  lines  of  induction  within  the  substance  of  the  conductor 
are  an  appreciable  proportion  of  the  total  number  of  lines  of  induc- 
tion which  link  the  conductor;  moreover, ^inc_e_.a_Jine  of  induction 
which  lies  wholly  within  the  substance  of  the  conductor  links  only 
that  portion  of  the  current  which  threads  the  loop  formed  by 
this  line,  it  follows  that  the  number  of  lines  of  induction  which 
link  the  inside  filaments  is  greater  than  the  number  which  link 
the  outside  or  surface  filaments,  and  consequently  the  back  elec- 
tromotive force  induced  in  the  inside  filaments  is  greater  than 
that  induced  in  the  outside  filaments.  The  result  of  this  is  that 
a  greater  proportion  of  the  current  flows  through  the  outside  fila- 
ments than  through  the  inside  filaments,  i.e.,  the  current  density 
is  greatest  near  the  surface  of  the  conductor.  In  certain  simple 
cases  the  exact  distribution  of  the  current  for  a  given  impressed 
electromotive  force  can  be  determined  by  expressing  the  condition 
that  the  impressed  electromotive  force  acting  on  each  filament 
must  be  equal  to  the  sum  of  the  resistance  drop  and  the  back 
electromotive  force  induced  in  that  filament.  This  phenomenon 
of  a  non-uniform  current  distribution  caused  by  a  rapid  variation 
of  the  current  with  respect  to  time  is  called  the  "  skin  effect."  As 
noted  in  Chapter  VII,  this  skin  effect  causes  an  increase  in  the  ap- 
parent resistance  of  the  circuit ;  the  self  inductance  of  a  circuit  is 
diminished  by  the  skin  effect. 


ELECTROMAGNETISM  223 

In  the  case  of  wires  of  the  size  ordinarily  employed  in  practice 
the  skin  effect  may  be  neglected  and  the  current  density  assumed 
to  be  constant  over  the  cross  section  of  the  wire  provided  the 
current  varies  comparatively  slowly  with  time.  (In  the  case  of 
an  alternating  current  the  frequency  may  be  as  high  as  60  cycles 
per  second  provided  the  conductor  is  non-magnetic ;  the  skin  effect 
in  a  steel  rail,  however,  is  quite  appreciable  even  for  low  fre- 
quencies.) The  assumption  of  no  skin  effect  is  equivalent  to  as- 
suming that  the  back  electromotive  force  induced  in  each  filament 
of  the  conductor  is  the  same,  and  therefore  that  the  number  of 
linkages  of  the  conductor  as  a  whole  may  be  taken  as  the  average 
of  the  linkages  of  all  the  filaments  taken  separately.  Approxi- 
mate formulas  for  self  inductance  may  also  be  obtained  by  neglect- 
ing altogether  the  lines  of  induction  inside  the  substance  of  the 
conductor. 

122.  Self  Inductance  of  Two  Long  Parallel  Wires.  —  As  an 
example  of  the  approximate  method  of  calculating  inductance, 
consider  a  circuit  formed  by  two  non-magnetic  parallel  wires, 
when  the  wires  are  so  long  that  the  magnetic  field  due  to  the  cur- 
rent in  the  conductors  connecting  their  ends  may  be  neglected. 


dx     D 


(Current  Down) 


-D-x 


(Current  Up) 


Fig.  73. 


Let  the  wires  be  of  the  same  size  and  have  a  radius  of  r  centi- 
meters, and  let  them  be  spaced  D  centimeters  between  centers. 
Let  the  current  in  each  wire  be  i  abamperes  ;  away  from  the  reader 
in  A  (Fig.  73)  and  toward  the  reader  in  B.  Then  the  field  in- 
tensity at  any  point  P  at  a  distance  x  from  the  center  of  the  wire 
A  on  the  line  between  A  and  B  is  (see  equation  4a  of  Chapter  III). 


x      D-x 

.Hence,  the  number  of  lines  of  induction  crossing  an  area  at   P 
of  width  dx  and  unit  length  parallel  to  the  two  wires  is,  since  the 


224  ELECTRICAL  ENGINEERING 

field  intensity  is  perpendicular  to  this  plane  and  the  permeability 
of  the  air  surrounding  the  wires  is  unity, 


Hence,  the  total  number  of  lines   of   induction  per   centimeter 
length  of  the  wires  threading  the  space  between  the  two  is 

D-r 


=  |      d<l>=2/      Inx-ln(D-x) 

J  x=r  _\x=r 


41  In 


(The  abbreviation  "In"  is  used  for  the  natural  logarithm  and 
the  abbreviation  "  log  "  for  the  common  logarithm.)  Whence 
the  inductance  of  one  centimeter  length  of  both  wires  is 

L  =—  =4  Tttf abhenries 

/  r 

Or,  since  each  wire  is  linked  by  half  the  resultant  lines  of  in- 
duction linking  the, space  between  the  two,  the  inductance  of 
each  wire  per  unit  length  is 

L  =2  In abhenries 

r 

These  formulas,  however,  are  only  approximate,  since  the  lines 
of  induction  inside  the  wires  have  been  neglected.  It  can  be 
shown  (see  Alex.  Russell,  Alternating  Currents,  Vol.  1,  p.  55)  that 
the  exact  formula  for  the  inductance  of  each  of  two  non-magnetic 
parallel  wires  for  slowly  varying  currents  is 

L=0.5+2  In  —  abhenries  per  cm.  (23a) 

r 


or 


or 


L  =0.01524  +  0.1403  log  —   millihenries  per  1000  ft.      (236) 

r 


L  =0.08047 +  0.7411  log—  millihenries  per  mile  (23c) 

r 

It  should  be  noted  that  since  D  and  r  occur  in  these  formulas  only 
as  a  ratio,  it  is  immaterial  in  what  units  D  and  r  be  expressed, 
provided  they  are  both  expressed  in  the  same  unit. 

The  minimum  value  of  the  inductance  is  when  the  wires  touch. 
In  this   case   D=2r,   and   therefore   L  =0.0575   millihenries   per 


ELECTROMAGNETISM  225 

1000  feet  and  is  independent  of  the  size  of  the  wire.  An  ab- 
solutely non-inductive  circuit  is  impossible,  although  this  condi- 
tion may  be  closely  approximated  by  placing  the  wires  forming 
the  circuit  close  together,  e.g.,  by  twisting  them  together. 

123.  Self  Inductance  of  a  Concentrated  Winding.  —  Let  Lx  be 
the  self  inductance  of  a  single  loop  of  wire  of  any  shape  ;  then  the 
flux  linking  this  loop  due  to  unit  current  in  it  is  <£  =Lt.     Consider 
a  coil  made  of   N  turns  of  wire,  each  turn  of  exactly  the  same 
dimensions  as  this  single  loop  and  let  these  N  turns  be  so  close 
together  that  they  may  all  be  considered  as  coinciding  exactly 
with  one  another,  i.e.,  let  the  N  turns  be  considered  as  concen- 
trated in  a  geometrical  line.     Then  unit  current  in  each  of  these 
turns  will  set  up  a  flux  <j>  which  will  link  each  of  the   N  turns  ; 
therefore  the  total  flux  linking  each  turn  will  be  N  <f>  =  NLl}  and 
since  there  are  N  turns  the  number  of  linkages  between  the  total 
flux  and  the  entire  coil  will  be  N2L^  which,  from  Article  117,  is 
equal  to  the  self  inductance  Ln  of  the  entire  coil,  i.e., 

Ln=Ar2L,  (24) 

Hence  the  self  inductance  of  a  concentrated  winding  is  proportional 
to  the  square  of  the  number  of  turns  in  it. 

124.  Self  Inductance  of  a  Long  Solenoid.  —  In  contradistinc- 
tion to  a  concentrated  winding,  consider  the  case  of  a  long  air-core 
solenoid,  which  is  one  form  of  a  distributed  winding.     Let  N  be 
the  number  of  turns  in  the  solenoid,  I  its  length  in  centimeters  and 
A   its   mean   cross   section   in  square   centimeters.     Then,  from 
equation  (4),  the  magnetic  field  intensity  at  any  point  inside  the 
solenoid  at  a  considerable  distance  from  its  ends  due  to  a  current 

of  one  abampere  in  the  coil  is  H  =  -    -  and  is  parallel  to  the 

axis  of  the  solenoid.  The  total  number  of  lines  of  induction  link- 
ing each  of  the  central  turns  (neglecting  the  lines  within  the  sub- 
stance of  the  wire)  is  therefore 


The  flux  linking  the  end  turns  is  less  than  this,  since  some  of  the 
lines  of  induction  go  through  the  lateral  walls  of  the  solenoid;  as 
an  approximation,  however,  in  the  case  of  a  long  solenoid  all  the 
lines  of  induction  may  be  assumed  to  link  these  end  turns  also. 
On  this  assumption  the  total  number  of  linkages  between  the  coil 


226  ELECTRICAL  ENGINEERING 

and  the  flux  per  unit  current  in  the  coil,  i.e.,  the  self  inductance  of 
the  coil,  is  then 


where  all  the  quantities  are  in  c.  g.  s.  units. 

The  self  inductance  of  a  solenoid,  therefore,  does  not  vary  with 
the  square  of  the  number  of  turns  when  the  variation  is  made  by 
changing  its  length,  for  in  this  case  the  number  of  turns  varies  as 
the  length,  and  therefore  the  inductance  varies'  directly  as  the 
number  of  turns,  or  directly  as  the  length.  Hence  when  a  sole- 
noid is  used  as  a  variable  inductance,  by  connecting  in  the  circuit 
a  greater  or  less  number  of  its  turns  by  means  of  a  sliding  contact, 
for  example,  the  inductance  varies  directly  as  the  distance  be- 
tween the  fixed  terminal  and  the  slider,  provided  this  distance  is 
large  compared  with  the  diameter  of  the  solenoid.  When  the 
slider  is  close  to  the  fixed  terminal,  the  turns  between  the  two 
form  an  approximately  concentrated  winding,  and  therefore  the 
inductance  when  only  a  small  part  of  the  winding  is  used  varies 
approximately  as  the  square  of  the  distance  between  the  two 
terminals. 

It  should  be  noted  that  when  a  coil  of  any  kind  is  used  'as  a 
part  of  an  electric  circuit,  there  is  always  a  mutual  inductance 
between  the  rest  of  the  circuit  and  the  coil.  This  mutual  induc- 
tance, however,  may  be  made  practically  negligible  by  making 
the  leads  to  the  coil  sufficiently  long  and  twisting  them  together. 

125.  Total  Energy  of  a  Magnetic  Field  in  Terms  of  the  Field 
Intensity  and  Flux  Density.  —  The  total  energy  of  the  magnetic 
field  due  to  any  number  of  electric  currents  may  also  be  expressed 
in  terms  of  the  field  intensity  and  the  flux  density.  Consider  the 
case  when  every  body  in  the  magnetic  field  has  a  constant  permea- 
bility; the  flux  density  and  field  intensity  at  every  point  in  the 
field  will  then  be  proportional  to-  each  other  and  in  the  same 
direction.  Under  these  conditions  the  energy  corresponding  to 
the  final  values  of  the  currents  must  be  independent  of  the  manner 
in  which  these  currents  are  established;  we  may  then  for  con- 
venience assume  that  the  currents  all  increase  proportionally  from 
zero  to  their  final  values.  Under  these  conditions,  if  we  imagine 
all  space  filled  with  tubes  drawn  in  such  a  manner  that  the  walls 
of  each  tube  are  tangent  at  each  point  to  the  direction  of  the  flux 
density  at  that  point  due  to  the  final  values  of  the  currents,  then 


ELECTROMAGNETISM  227 

the  walls  of  these  tubes  will  also  be  tangent  to  the  flux  density 
at  any  instant  while  the  currents  are  rising  from  zero  to  their  final 
values.  These  tubes  will  also  be  closed  on  themselves  and  from 
equation  (8)  the  current  linked  by  each  tube  at  any  instant  will 


be  i  =  7—  I   Hdl,  where   dl  is   any  elementary   length  measured 

71 J  \L\ 

along  the  axis  of  the  tube  and  H  the  intensity  of  the  field  at  dl. 
When  all  the  currents  change  by  a  certain  small  amount,  producing 
a  change  of  flux  density  dB  at  every  point  in  the  field,  the  change 
in  the  flux  through  each  tube  will  be  d  <f>  =dB.ds  where  ds  repre- 
sents any  section  of  the  tube  at  right  angles  to  its  axis,  and  dB 
the  change  in  the  flux  density  at  this  point.  dB  and  ds  will  of 
course  vary  from  point  to  point  in  the  field  but,  from  the  manner 
in  which  these  tubes  are  drawn,  the  product  dB.ds  will  be  constant 
for  every  section  of  any  one  tube.  The  work  done  in  changing  the 
flux  through  each  tube  by  an  amount  d  </>  is 


=  -—  dB.ds   I 
4?r 

J    \ 


Hdl 

\L\ 

and  the  total  work  done  corresponding  to  all  the  tubes  is  the 
volume  integral 


—    I  dB.ds  \  Hdl 
4ir 

J   \S\          J    |L| 


throughout   all  space.     But,   since  dB.ds  is  constant  along  any 
tube,  this  may  be  written 


-   I         I      HdBds.dl=-~    I 

t  77   I  I  47T     I 

J   \S\  J   \L\  J    \ 


HdBdv 
\v\ 

since  ds.dl  represents  an  elementary  volume  dv  at  each  point. 
Hence  the  work  done  per  unit  volume  of  space,  when  the  currents 
in  the  various  circuits  change  proportionally  by  an  infinitesimal 
amount,  is 

dw=—Hd.B 

47T 

Hence  the  total  work  done  per  unit  volume  of  space  by  any  number 
of  currents  in  establishing  their  resultant  magnetic  field  is 

HdB=jirz=^-  (26) 


provided  the  permeability  of  every  body  in  the  field  is  constant. 


228  ELECTRICAL  ENGINEERING 

When  the  permeability  of  the  bodies  in  the  field  is  not  con- 
stant, but  varies  with  the  field  intensity,  the  reasoning  employed 

i  rB 

in  deducing  the  expression  —  I      HdB  for  the  work  done   by 

the  currents  per  unit  volume  of  the  field,  breaks  down ;  for  in  this 
case  the  direction  of  the  tubes  which  we  considered  as  filling  all 
space  may  change  as  the  currents  change  in  value.  However,  the 


expression  - —  I      HdB  for  the  energy  per  unit  volume  required 

J  o 
to  establish  the  field  (only  a  portion  of  this,  however,  is  stored  in 

the  field,  see  Article  126a)  is  applicable  to  any  case  where  the 
direction  of  the  lines  of  induction  remains  unaltered  and  coincides 
with,  or  is  in  the  opposite  direction  to,  the  lines  of  force.  This 
condition  is  realized  in  the  special  case  of  a  closed  anchor  ring 
which  is  magnetised  by  a  coil  uniformly  wound  around  it  (as 
described  in  Article  110)  and  is  also  approximately  realized  in  the 
magnetic  circuits  of  most  electrical  apparatus. 

126.  Heat  Energy  Due  to  Hysteresis.  —  When  the  permeability 
of  every  body  in  the  magnetic  field  is  constant,  the  energy  trans- 
ferred to  the  magnetic  field  when  the  flux  density  increases  from 
zero  to  any  value  B  is  exactly  equal  to  the  energy  transferred  from 
the  magnetic  field  to  the  electric  currents  when  the  flux  density 
decreases  from  B  to  zero;  for,  when  B  is  proportional  to  H, 

i  CB  if 

HdB  =  —  —   I        Hd  B       However,  in  the  case  of  ordi- 
4wJo  47J* 

nary  magnetic  bodies  such  as  iron  or  steel,  we  have  already  seen 
(Article  57)  that  the  relation  between  B  and  H  when  the  flux 
density  increases  is  different  from  the  relation  between  these  two 
quantities  when  the  flux  density  decreases.  Hence  in  this  case 

i  cB  i  r 

—  I      HdB  is  not  equal  to HdB.     The  first  expres- 

J    o  ^  J    B 

sion,  however,  represents  the  work  done  by  the  currents  in  estab- 
lishing themselves,  while  the  second  expression  represents  the 
work  done  on  the  currents  when  the  magnetic  field  disappears; 
the  difference  between  these  two  expressions  must  then  represent 
the  production  of  some  other  form  of  energy  in  the  magnetic  sub- 
stance, and  experiment  shows  that  this  other  form  of  energy  is 
heat  energy.  That  is,  whenever  the  flux  density  in  a  piece  of  any 


ELECTROMAGNETISM  229 

ordinary  magnetic  substance  is  changed,  heat  energy  is  always 
developed.  Consequently,  when  the  flux  density  in  a  piece  of 
iron  or  steel  is  changed  from  any  value  B  to  any  other  value  and 
then  brought  back  again  to  the  original  value  B,  an  amount  of 
heat  energy  per  unit  volume  of  the  substance  equal  to 


w= —  I  HdB  (27) 

47T   I 

J    \B\ 

for  this  complete  cycle  of  values,  is  produced  in  the  substance. 
Hence,  when  this  cycle  of  values  is  plotted  (with  H  and  B  to  the 
same  scale)  in  a  curve  (i.e.,  the  hysteresis  curve),  the  area  of  this 
curve  divided  by  4  TT  gives  the  amount  of  heat  energy  per  unit 
volume  "  due  to  hysteresis  "  produced  in  the  substance.  When 
H  and  B  are  expressed  in  c.  g.  s.  electromagnetic  units,  this  heat 
energy  is  in  ergs  per  cubic  centimeter. 

126a.  Tractive  Force  of  an  Electromagnet.  —  From  the  formula, 
equation  (26),  for  the  energy  per  unit  volume  of  a  magnetic  field, 
can  be  deduced  directly  the  force  of  attraction  between  two  elec- 
tromagnets or  between  an  electromagnet  and  its  keeper.  Con- 
sider the  special  case  where  the  surface  of  separation  of  the  two 
parts  of  the  magnetic  circuit  is  perpendicular  to  the  lines  of  in- 
duction; let  S  be  the  area  of  this  surface  in  square  centimeters 
and  let  dx  be  the  infinitesimal  amount  by  which  the  two  parts  of 
the  magnetic  circuit  are  separated.  When  dx  is  infinitely  small, 
the  change  produced  in  the  reluctance  of  the  circuit  is  negligible, 
and  therefore  the  flux  density  in  the  air  gap  formed  is  the  same 
as  originally  existed  in  the  iron ;  let  this  flux  density  be  B  and  let 
it  be  assumed  constant  over  the  surface  of  separation.  The 
volume  of  the  magnetic  field  is  changed  by  an  amount  Sdx,  and 
consequently  the  energy  of  the  magnetic  field,  from  equation  (26), 
is  changed  by  the  amount 

B2 

=8^r  C 

But  the  work  done  in  separating  the  two  parts  of  the  circuit  is 
also  equal  to  the  product  of  the  displacement  dx,  and  the  mechani- 
cal force  F  in  dynes  required  to  produce  the  separation,  i.e., 

dW  =  Fdx 
Equating  these  two  expressions,  we  have 

F  = 

STT  (28) 


ELECTRICAL  ENGINEERING 

rhen  the  flux  density  is  not  constant  over  the  surface  of  separa 
/on,  the  total  force  required  is  the  surface  integral 

(28a) 


SUMMARY  OF  IMPORTANT  DEFINITIONS  AND 
PRINCIPLES 

1.  The  number  of  linkages  between  a  line  of  induction  and  a 
coil  of  wire  is  equal  to  the  number  of  turns  of  this  coil  linked  by 
the  line  of  induction.     The  total  number  of  linkages  between  a 
coil  and  the  flux  threading  it  is  the  sum  of  the  linkages  of  all  the 
lines  of  induction.     For  a  concentrated  winding  of  N  turns  and 
each  turn  linked  by  <f>  lines,  the  number  of  linkages  is 

\=N<f> 
Linkages  are  expressed  in  mawells. 

2.  Whenever  the  flux  threading  a  circuit  changes  an  electro- 
motive force  is  induced  in  this  circuit  equal  to  the  time  rate  of 
change  of  the  linkages  between  the  flux  and  the.  circuit,  i.e., 

e=  —  —  ab  volts  =  — 10'8 —  volts 

dt  dt 

where  X  is  in  maxwells  and  t  in  seconds.  The  minus  sign  in  the 
above  equation  indicates  that  the  direction  of  the  electromotive 
force  resulting  from  an  increase  in  the  flux  in  the  same  direction 
as  that  of  the  flux  due  to  the  current  in  the  circuit  produces  an 
electromotive  force  in  the  opposite  direction  to  that  of  the  current. 

3.  The  work  done  on  an  electric  current  when  the  number  of 
linkages  between  the  circuit  and  the  flux  linking  it  increases  by 
an  amount  d\  is 

dW~  —  id\  ergs 

where  i  is  in  abamperes  and  X  in  maxwells. 

4.  When  the  change  in  the  linkages  between  an  electric  circuit 
and  the  flux  linking  it  is  caused  by  the  motion  of  a  part  of  this 
circuit  (e.g.  a  wire)  through  a  magnetic  field,  the  induced  electro- 
motive force  is  equal  to  the  rate  at  which  this  part  of  the  circuit 
cuts  the  lines  of  induction.     The  direction  of  the  electromotive 
force  is  the  direction  in  which  the  middle  finger  of  the  right  hand 
points  when  laid  along  the  wire  and  the  thumb  and  forefinger 
of   this   hand  are  held  perpendicular  to  each  other  and  to  the 
middle  finger,  with  the  thumb  pointing  in  the  direction  of  the 


ELECTROMAGNETISM  231 

motion  of  the  wire  and  the  forefinger  in  the  direction  of  the  flux 
density.  The  value  of  the  electromotive  force  induced  in  a  wire 
I  centimeters  long  when  it  moves  perpendicularly  across  a  mag- 
netic field  of  flux  density  of  B  gausses  with  a  velocity  of  v  centi- 
meters per  second  perpendicular  to  itself  is 
e  =  Blv  abvolts 

5.  The  intensity  of  the  magnetic  field  due  to  a  current  of  i 
abamperes  in  a  long  solenoid  of  A/"/'  turns  per  centimeter  length  is 

H  =  4TrN'i  gilberts  per  cm      Q~[      l^~~f /jV    ^ 

6.  The  quantity  of  electricity  discharged  through  a  circurtrof~?T 
abohms  resistance  when  the  number  of  linkages  between  the 
circuit  and  the  flux  threading  it  changes  from  \v  to  X2  maxwells  is 

Q  =  — abcoulombs 

R 

7.  The  average  value  of  the  induced  electromotive  force  be- 
tween the  brushes  of  a  continuous  current  dynamo  is 

_.     npfb  N 

E  =  J-L—  volts 

p'XlO8 

where  n  is  the  number  of  revolutions  per  second,  p  the  number  of 
field  poles,  p'  the  number  of  parallel  paths  through  the  armature, 
<j)  the  useful  flux  per  pole  in  maxwells,  and  N  is  the  number  of 
conductors  which  cut  the  flux. 

8.  The  work  required  to  carry  a  unit  north  pole  around  a 
closed  path  is  called  the  magnetomotive  force  acting  around  this 
path.     The  magnetomotive  force  acting  around  a  closed  path 
which  links   N  turns  of  a  circuit  in  which  the  current  is  i  ab- 
amperes is 

4  TT  N  i  gilberts 

The  product  of  the  current  in  amperes  by  the  number  of  turns  is 
called  the  ampere-turns  of  the  coil. 

1  gilbert  =0.79578  ampere-turn 

9.  The  reluctance  of  a  given  portion  of  a  magnetic  circuit  is 
defined  as  the  ratio  of  the  drop  of  magnetic  potential  along  the 
lines  of  induction  to  the  total  number  of  lines  of  induction  through 
the  given  portion  of  the  circuit.     The  c.  g.  s.  electromagnetic  unit 
of  reluctance  is  the  oersted.     The  reluctance  of  a  portion  of  a 
circuit  /  centimeters  long,  A  square  centimeters  in  cross  section 
and  of  a  permeability  p  is 

R=  oersteds 


232  ELECTRICAL   ENGINEERING 

provided  the  flux  density  is  uniform  and  perpendicular  to  the  cross 
section. 

10.  The  ampere-turns  required  to  establish  a  given  flux  are 
found  by  equating  the  drop  of  magnetic  potential  through  the 
magnetic  circuit  to  the  magnetomotive  force,  i.e., 


where  H  is  in  ampere-turns  per  inch,  I  is  in  inches,  7  in  amperes, 
and  the  integral  is  taken  around  the  closed  path  formed  by  the 
magnetic  circuit. 

11.  The  value  of  the  ratio  of  the  electromotive  force  induced 
in  an  electric  circuit,  due  solely  to  a  change  of  the  current  in  this  cir- 
cuit, to  the  time  rate  of  this  change,  is  called  the  self  inductance 
L  of  this  circuit  ;  hence  the  self-induced  electromotive  force  is 

Tdi 
e  =  L  — 

dt 

The  c.  g.  s.  unit  of  self  inductance  is  the  abhenry  ;  the  practical  unit 
is  the  henry. 

1  henry  =109  abhenries 

When  the  permeability  of  every  body  in  the  magnetic  field  is 
constant  and  the  circuit  remains  unaltered  in  shape,  the  self  in- 
ductance is  a  constant  of  the  circuit  (for  a  given  distribution  of 
current)  equal  to  the  number  of  linkages  between  this  circuit  and 
the  flux  produced  by  unit  current  in  it. 

12.  The  value  of  the  ratio  of  the  electromotive  force  induced 
in  an  electric  circuit  1,  due  to  a  change  of  the  current  in  any  other 
circuit  2,  to  the  time  rate  of  change  of  this  current,  is  called  the 
mutual  inductance  M  12  of  circuit  2  with  respect  to  circuit  1  ;  hence 
the  electromotive  force  induced  in  1  due  to  a  change  of  the  cur- 
rent ia  in  2  is 

612=  M12  -^ 
dt 

The  units  of  mutual  inductance  are  the  same  as  the  units  of  self 
inductance.  When  the  permeability  of  every  body  in  the  mag- 
netic field  is  constant  and  the  circuits  are  fixed  with  respect  to 
each  other  and  remain  unaltered  in  size  and  shape,  the  mutual 
inductance  of  one  circuit  with  respect  to  the  other  is  constant  (for 
a  given  distribution  of  the  currents)  and  equal  to  the  number  of 
linkages  between  one  circuit  and  the  flux  produced  by  unit  current 
in  the  other,  and  the  mutual  inductance  of  the  one  circuit  with 


ELECTROMAGNETISM  233 

respect  to  the  other  is  the  same  as  the  mutual  inductance  of  the 
second  circuit  with  respect  to  the  first. 

13.  The  energy  of  the  magnetic  field  due  to  a  current  i  in  a 
circuit  of  constant  self  inductance  L  is 

W  =  \L? 

When  L  is  in  henries  and  i  in  amperes  this  formula  gives  the  energy 
in  joules  ;  when  L  is  in  abhenries  and  i  in  abamperes  this  formula 
gives  the  energy  in  ergs. 

14.  The  energy  of  the  magnetic  field  due  to  the  currents  t\  and 
i2  in  two  circuits  which  have  constant  self  inductances  Lv  and  L2 
and  a  constant  mutual  inductance  M  is 


When  the  inductances  are  in  henries  and  the  currents  in  amperes 
this  formula  gives  the  energy  in  joules  ;  when  these  quantities  are 
in  abhenries  and  abamperes  this  formula  gives  the  energy  in  ergs. 

15.  When  the  current  in  a  conductor  of  large  cross  section 
varies  rapidly  with  time,  the  back  electromotive  force  due  to  the 
variation  of  the  flux  linking  the  conductor  is  considerably  greater 
in  the  inside  filaments  of  the  conductor  than  in  the  outside  fila- 
ments ;  consequently  the  current  density  is  greatest  near  the  sur- 
face of  the  conductor.     This   phenomenon  is  known  as  the  skin 
effect. 

16.  The  self  inductance  of   each  of   two  long  parallel  non- 
magnetic wires,  assuming  a  uniform  current  density,  is 

L  =0.01524  +  0.1403  log  —  millihenries  per  mile 

r 

where  D  is  the  distance  between  centers  and  r  the  radius  of  each 
wire,  both  in  the  same  units. 

17.  The   self   inductance    of    a    concentrated    winding    varies 
directly  as  the  square  of  the  number  of  turns. 

18.  The  self  inductance  of  a  long  solenoid  is  approximately 

47T7VM 
L  =  --  abhenries 

where  N  is  the  total  number  of  turns  in  the  solenoid,  A  its  cross 
section  in  square  centimeters  and  I  its  length  in  centimeters. 

19.  The  energy  per  cubic  centimeter  of  any  magnetic  field 
when  the  permeability  of  every  body  in  the  field  is  constant  is 


8?r 


ergs 


234  ELECTRICAL  ENGINEERING 

where  H  is  the  field  intensity  in  gilberts  per  centimeter,  B  the 
flux  density  in  gausses  and  /*  the  permeability  in  c.  g.  s.  electro- 
magnetic units. 

20.  The  heat  energy  dissipated  per  cycle  per  centimeter- cube 
in  a  magnetic  substance  when  an  alternating  magnetic  field  is 
established  in  it  is  equal  to  the  area  of  the  corresponding  hysteresis 
loop  divided  by  4  TT. 

21.  The  force  required  to  separate  two  parts  of  a  magnetic 

circuit  is  D2  A 

»**•"•  i 

F= — —  dynes 

STT 

where  A  is  the  area  in  sq.  cm.  of  the  surface  of  separation  and  B 
the  flux  density  in  gausses  normal  to  this  surface,  provided  B  is 
constant. 

PROBLEMS 

1.  A  coil  which  has  a  concentrated  winding  of  100  turns  is 
revolved  in  a  uniform  magnetic  field  about  an  axis  perpendicular 
to  the  direction  of  the  field.     The  intensity  of  the  field  is  100 
gilberts  per  cm.,  the  area  of  each  turn  of  the  coil  is  10  sq.  cm. ; 
the  coil  is  rotated  with  a  uniform  velocity  of  25  revolutions  per 
second.     What  is  the  instantaneous  value  of  the  e.  m.  f.  induced 
in  the  coil,  (1)  when  its  plane  is  perpendicular  to  the  field,  (2)  when 
its  plane  is  parallel  to  the  field;  (3)  what  is  the  average  value  of 
the  e.  m.  /.  induced  in  the  coil  while  it  is  rotating  180°  from  the 
first  position? 

Ans.:  (1)  0;    (2)  1.013  volts;    (3)  0.645  volts. 

2.  How  many  foot-pounds  of  work  must  be  done  to  pull  a 
coil  having  a  concentrated  winding  of  200  turns  out  of  a  magnetic 
field,  if  the  flux  threading  the  coil  is  100,000  maxwells  and  the 
current  in  it  is  maintained  constant  at  100  amperes? 

Ans.:  14.76  foot-pounds. 

3.  A  straight  wire  10  inches  long  is  moved  parallel  to  itself 
across  a  magnetic  field  in  a  plane  making  an  angle  of  45°  with  the 
direction  of  the  field.     What  is  the  value  of  the  e.  m.  f.  in  volts 
induced  in  the  wire  if  it  is  moved  with  a  velocity  of  5  feet  per 
second  and  the  field  intensity  is  200  gilberts  per  cm.? 

Ans.:  0.00548  volts. 

4.  A  coil  of  20  turns  is  wound  over  the  middle  of  a  long  solenoid 
with  an  air  core.     The  solenoid  has  a  cross-sectional  area  of  10 
sq.  cm.  and  has  50  turns  per  linear  inch.     If  the  20-turn  secondary 


ELECTROMAGNETISM  235 

is  connected  in  series  with  a  resistance  such  that  the  total  resist- 
ance of  the  secondary  circuit  is   1000  ohms,  what  quantity   of 
electricity  in  coulombs  is  discharged  through  the  secondary  circuit 
when  a  current  of  5  amperes  is  reversed  in  the  primary? 
Ans.:  4.95X10'7  coulombs. 

5.  A  coil  of  1000  turns  is  wound  uniformly  on  a  round  iron  bar 
20  centimeters  in  length  and  2  sq.  cm.  in  cross  section.     If  a  con- 
stant current  of  5  amperes  is  established  in  the  coil,  calculate  the 
work  in  joules  required  to  withdraw  the  bar  from  the  coil,  assuming 
the  permeability  of  the  bar  to  be  constant  and  equal  to  400  c.  g.  s. 
units  and  neglecting  the  demagnetising  effect  of  the  ends  of  the  bar. 

Ans.:  0.01252  joules. 

6.  An  air-core  solenoid  of  1000  turns  is  20  cm.  long  and  has  a 
mean  diameter  of  5  cm.     Calculate  the  field  intensity  on  the  axis 
of  the  solenoid  5  cm.  internally  from  one  end,  due  to  a  current  of 
2  amperes  in  the  solenoid;  make  this  calculation  by  integrating 
the  effect  due  to  each  turn  separately.     What  is  the  error  involved 
in  using  formula  (4)  for  the  field  intensity  at  this  point? 

Ans.:  122.8    gilberts    per    cm.     Formula  (4)    gives    a    value 
20.8%  too  great. 

7.  A  cast-iron  ring  20  inches  in  mean  diameter  with  a  circular 
cross  section  2  inches  in  diameter  has  an  air  gap  0.1  inch  in  length. 
(1)  How  many  ampere-turns  are  required  to  produce  an  average 
flux  density  of  5000  gausses  in  this  air  gap,  assuming  no  leakage? 
Use  Fig.  38  to  get  the  relation  between  flux  density  and  field 
intensity  in  the  iron.     (2)  What  is  the  percentage  ratio  of  the 
drop  of  magnetic  potential  in  the  gap  to  that  in  the  iron? 

Ans.:  (1)  4090  ampere-turns;  (2)  32.8%. 

8.  What  is  the  total  reluctance  of  the  iron  ring  and  air  gap 
described  in  Problem  7? 

Ans.:  0.159  oersteds. 

9.  An  iron  ring  20  cm.  in  mean  diameter  and  4  sq.  cm.  in  cross 
section  is  wound  uniformly  with  2000  turns  of  wire.     Determine 
the  self  inductance  of  this  winding  in  henries,  assuming  a  constant 
permeability  of  500  c.  g.  s.  units. 

Ans.:  1.6  henries. 

10.  Two  coils  A  and  B  are  wound  uniformly  upon  the  same 
magnetic  circuit,  one  over  the  other.     The  self  inductance  of  A 
is  3  henries  and  the  self  inductance  of  B  is  5  henries.     Assuming 
the  permeability  of  the  magnetic  circuit  to  be  constant,  determine 


236  ELECTRICAL  ENGINEERING 

the  total  inductance  of  A  and  B  when  connected  in  series,  ( 1)  so 
that  their  magnetic  fields  are  in  the  same  direction,  and  (2)  so  that 
their  magnetic  fields  are  in  opposite  directions.  (3)  What  is  the 
mutual  inductance  of  A  and  B? 

Ans.:  (1)  15.74  henries;   (2)  0.26  henries;  (3)  3.87  henries. 

11.  A  coil  having  a  concentrated  winding  of  1000  turns  has  a 
self  inductance  of  0.05  henries.     What  is  the  energy  in  joules  of 
the  magnetic  field  due  to  the  current  in  the  qoil  when  this  current 
produces  a  flux  of  2000  maxwells? 

Ans.:  0.004  joule. 

12.  When  a  current  of  4  .amperes  is  established  in  the  field 
winding  of  a  4-pole  shunt  generator,  the  flux  of  induction  through 
each  field  coil  is  3X106  maxwells.     The  four  field  coils  are  con- 
nected in  series  electrically  and  each  has  800  turns.     Assuming 
constant  permeability  and  no   magnetic   leakage,   determine  (1) 
the  self  inductance  of  the  entire  field  winding,  and  (2)  the  total 
energy  of  the  magnetic  field  of  the  generator. 

Ans.:  (1)  24  henries;  (2)  384  joules. 

13.  A  cast-iron  ring  10  inches  in  mean  diameter  and  4  sq.  in. 
in  cross  section  is  divided  into  two  halves.     Upon  each  half  of  the 
ring  are  wound  uniformly  100  turns  of  wire.     If  the  two  halves  of 
the  ring  are  placed  together  to  form  a  complete  ring  and  the  two 
coils  are  connected  in  series  so  that  their  magnetic  fields  add, 
determine  ( 1)  the  energy  of  the  magnetic  field  per  cubic  centimeter 
when  the  current  established  in  the  winding  is  2  amperes,  assuming 
a  constant  permeability  of  300  c.  g.  s.  units,  and  (2)  the  initial  force 
in  pounds  required  to  pull  the  two  halves  of  the  ring  apart. 

Ans.:  (1)  473  ergs;  (2)  16.5  pounds. 


V 
ELECTROSTATICS 

127.  Electric  Charges.  —  We  have  already  seen  (Article  64) 
that  it  is  possible  to  have  a  difference  of  electric  potential  between 
two  conductors  without-  having  an  electric  current  in  these  con- 
ductors. For  example,  the  difference  of  electric  potential  between 
the  two  poles  of  a  battery  when  there  is  no  conductor  connecting 
its  poles,  that  is,  when  the  battery  is  "  open-circuited,"  is  equal 
to  the  electromotive  force  of  the  battery.  It  is  found  by  ex- 
periment that  when  a  conductor  A  of  any  kind  is  connected  to  one 
of  the  poles  of  a  battery  but  not  to  the  other,  a  momentary  electric 
current  is  established  in  this  conductor,  but  after  a  small  fraction 
of  a  second  this  current  ceases.  By  Ohm's  Law,  every  point  of 
this  conductor  must  come  to  the  same  potential  as  the  terminal 
of  the  battery  to  which  it  is  connected  (neglecting  any  slight 
contact  electromotive  force  between  the  conductor  and  the 
terminal).  Similarly,  in  any  other  conductor  B  connected  to 
the  other  terminal  of  the  battery  but  completely  insulated  from 
the  first  terminal,  a  momentary  electric  current  will  be  established, 
but  when  this  current  ceases  every  point  of  the  conductor  B 
will  be  at  the  same  potential  as  that  of  the  terminal  to  which 
it  is  connected.  Hence  between  the  two  conductors  A  and  B  a 
difference  of  potential  is  established  equal  to  that  of  the  electro- 
motive force  of  the  battery.  It  is  also  found  by  experiment, 
that  the  wires  connecting  the  two  conductors  A  and  B  to  the 
battery  may  be  removed  without  changing  this  difference  of 
electric  potential  between  them,  provided  the  wires  are  small 
and  the  position  of  the  conductors  A  and  B  relative  to  each 
other  and  all  other  bodies  in  their  vicinity  remains  fixed  and  the 
conductors  remain  perfectly  insulated  from  each  other  and  all 
other  conductors.  This  can  be  tested  by  again  connecting  the 
conductors  A  and  B  to  the  same  terminals  of  the  battery ;  it  will 
be  found  that  no  current,  not  even  a  momentary  current,  is  estab- 
lished in  the  wires,  thus  showing  that  each  conductor  remains 
at  the  potential  of  the  terminal  to  which  it  was  originally  con- 

237 


238  ELECTRICAL  ENGINEERING 

nected.  A  difference  of  electric  potential  may  then  exist  between 
two  conductors  which  are  entirely  insulated  from  each  other 
and  all  other  conductors. 

It  is  found  by  experiment  that  whenever  a  difference  of 
electric  potential  is  established  between  any  two  conductors 
these  conductors  exert  a  force  upon  each  other  and  in  general 
also  upon  all  other  conductors  and  dielectrics  in  their  vicinity, 
even  when  there  are  no  currents  in  the  conductors.  These  forces 
are  due  solely  to  the  difference  of  electric  potential  established 
between  the  various  bodies,  and  as  long  as  these  potential  differ- 
ences remain  constant  in  value  the  forces  are  found  to  remain 
constant.  Just  as  the  forces  produced  by  magnets  on  one  another 
and  upon  magnetic  substances  may  be  expressed  in  terms  of  a 
something  associated  with  the  surfaces  of  the  magnets  and  mag- 
netic bodies,  so  may  these  forces  due  to  differences  of  electric 
potential  be  expressed  in  terms  of  a  something  associated  with 
the  surfaces  of  the  conductors  and  the  dielectrics  in  their  vicinity. 
The  something  that  produces  these  forces  which  are  due  to  differ- 
ences of  electric  potential  is  called  an  electric  charge.  We  wish 
now  to  see  what  are  the  properties  of  these  electric  charges  and 
what  is  their  relation  to  an  electric  current. 

In  the  first  place,  it  is  found  that  forces  of  exactly  the  same 
nature  are  produced  between  any  two  dielectrics,  such  as  glass 
and  silk,  when  these  two  bodies  are  rubbed  together  and  then 
separated;  also  when  an  insulated  conductor  is  rubbed  with  any 
dielectric,  such  as  a  piece  of  silk,  and  the  conductor  and  the 
dielectric  are  separated,  it  is  found  that  the  two  bodies  attract 
each  other.  Bodies  which  are  charged  in  this  way  are  said  to  be 
"  charged  by  friction."  In  fact,  this  method  of  producing  the 
phenomena  which  are  ascribed  to  electric  charges  was  known  to 
the  ancients ;  whereas  batteries  and  generators  are  comparatively 
recent  inventions. 

128.  Positive  and  Negative  Charges.  —  Attraction  and  Re- 
pulsion of  Charged  Bodies.  —  It  is  found  by  experiment  that 
bodies  which  are  charged  in  exactly  the  same  manner,  as  for 
example,  two  insulated  conductors  placed  momentarily  in  con- 
tact with  the  same  terminal  of  a  generator,  always  repel  each 
other,  while  they  may  repel  or  attract  a  body  which  has  been 
charged  in  some  other  manner.  For  example,  an  insulated 
conductor  A  placed  momentarily  in  contact  with  the  positive 


ELECTROSTATICS  239 

terminal  of  a  generator  is  found  to  repel  a  body  B  which  has 
been  placed  momentarily  in  contact  with  this  terminal,  but  will 
attract  the  body  B  if  the  latter  is  placed  in  contact  with  the 
negative  terminal  of  the  generator.     (These  forces  are  perceptible 
only  in  case  the  generator  develops  an  e.  m.  /.  of  several  thousand 
volts  or  more.)     Similarly,  two  pieces  of  glass  which  have  been 
rubbed  with  silk  are  found  to  repel  each  other,  while  the  silk 
and  glass  are  found  to  attract  each  other.     We  are  therefore  led 
to  the  conception  of  two  kinds  of  electric   charges,  which  are 
called  respectively  positive  and  negative  charges.     A  body  which 
repels  a  piece  of  glass  which  has  been  rubbed  with  silk  is  said  to 
be  positively  charged,  or  to  "  have  "  a  positive  charge,  while  a 
body  which  attracts  a  piece  of  glass  which  has  been  rubbed  with 
silk  is  said  to  be  negatively  charged  or  to  "  have  "  a  negative  charge. 
A  conductor  which  is  placed  in  contact  with  the  positive  pole 
of  a  battery  or  other  source  of  electromotive  force  is  positively 
charged,  while  a  body  placed  in  contact  with  the  negative  pole  of  a 
battery  or  other  source  of  electromotive  force  is  negatively  charged. 
129.  Charging  by  Contact  and    by  Induction.  —  It  is  found 
by  experiment  that  when  an  originally  uncharged  conductor    U 
(e.g.,  a  solid  piece  of  metal,  not  necessarily  a  wire)  is  placed  in 
contact  with  a  charged  conductor  C,  the  uncharged  conductor 
likewise  receives   a  charge    which  is   of  the  same  sign  as  that 
of  the  charged  conductor;  that  is,  when  the  charged  conductor 
C  is   positively  charged,  the   originally  uncharged  conductor   U 
placed  in  contact  with  it  likewise  becomes  positively  charged; 
while  if  the  originally  uncharged  conductor  U  is  placed  in  contact 
with  a  negatively  charged  conductor  C" ',  the  uncharged  conductor 
U   becomes    negatively    charged.      When    the    conductor    U   is 
removed  from  contact  with  the  charged  conductor  it  is  found 
that  it  remains  charged,  provided  it  is  perfectly  insulated  from  all 
other  conductors.     A  conductor  which  is  thus  charged  by  being 
placed  in  contact  with  another  conductor  is  said  to  be  "  charged 
by  contact."* 

*It  is  also  possible  to  charge  the  surface  of  a  dielectric  by  placing  it  in 
contact  with  a  charged  conductor  or  a  charged  dielectric.  Very  little  is  known 
concerning  the  exact  distribution  of  such  charges  on  dielectrics.  Fortunately, 
where  a  charged  conductor  is  in  contact  with  a  dielectric  it  is  in  general 
immaterial  whether  the  charge  is  considered  as  "residing"  on  the  surface 
of  the  conductor  or  on  the  surface  of  the  dielectric  or  on  both.  These  contact 
charges  on  dielectrics  must  not  be  confused  with  the  induced  charges  dis- 
cussed later. 


240  ELECTRICAL  ENGINEERING 

It  is  also  found  by  experiment  that  a  conductor  can  be  charged 
without  placing  it  in  actual  contact  with  a  charged  conductor; 
merely   placing   the   originally   uncharged   conductor     U   in   the 
vicinity  of  a  charged  body  C  will  cause  the  originally  uncharged 
conductor    U  to  become  charged.     In  this  case,  however,  when 
the  originally  uncharged  conductor   U  is  perfectly  insulated,  it  is 
found  that  the  portion  of  this  conductor   U  nearer  the  charged 
body  C  receives  a  charge  of  the  opposite  sign  to  that  on  C  while 
the  more  remote  portion  of  the  conductor   U  receives  a  charge  of 
the  same  sign  as  that  of  C.     In  other  words,  the  conductor    U 
becomes  charged  by  "  induction  "  in  a  manner  similar  to  that 
by  which  a  piece  of  soft  iron  placed  in  a  magnetic  field  becomes 
magnetised    by    induction.     There    is,    however,    an    important 
difference  between  the  phenomenon  of  electrostatic  induction  in 
a  conductor  and  the  phenomenon  of  magnetic  induction  in  iron. 
When  a  piece  of  iron  which  is  magnetised  by  induction  is  divided 
into  two  parts  and  the  two  parts  are  separated  slightly,  so  that  a 
narrow  gap  is  formed  between  them,  magnetic  poles  are  found 
to  exist  on  the  surfaces  of  the  iron  forming  the  walls  of  this  gap. 
When  a  conductor  U  (which  may  be  made  in  two  parts   originally 
in  contact)  is  charged  by  induction  and  the  two  parts    Ul  and 
t/,  are  separated  a  slight  distance,  the  surfaces  of    Ul  and    U2 
forming  the  walls  of  this  gap  are  not  charged.     Again,  when  the 
piece  of  iron  which  is  magnetised  by  induction  is  separated  into 
two  parts  and  either  part  is  removed  from  the  magnetic  field, 
this  portion  of  the  original  piece  of  iron  is  found  to  have  equal 
and  opposite  poles  which   disappear  almost  entirely  when   the 
iron  is  jarred;  however,  when  a  conductor  which  is  charged  by 
induction  is  separated  into  two  parts   U1  and   U2,  it  is  found  that 
the  two  portions  of  the  conductor  retain  their  charges,  one  part 
a  positive  charge  and  the  other  part  a  negative  charge,  as  long  as 
the  two  parts  are  kept  perfectly  insulated  from  each  other  and 
from  all  other  conductors.     The  charges  on  the  two  portions  of 
the  conductor  also  remain  unaltered  in  amount  (but  not  in  dis- 
tribution, as  we  shall  see  later)  even  when  these  two  portions 
are  removed  from  the  vicinity  of  the  conductor  which  induced 
the  charges  and  are  separated  from  each  other  by  any  distance. 
Hence   another  important   difference   between   the   phenomenon 
of  electrostatic  induction  in  a  conductor  and  the  phenomenon 
of  magnetic  induction;  the  total  strength  of  the  poles  on  a  mag- 


ELECTROSTATICS  241 

netic  substance  is  always  zero,  no  matter  into  how  many  pieces 
the  substance  may  be  broken,  but  the  positive  and  negative 
charges  induced  on  a  conductor  may  be  separated  from  each  other 
by  dividing  the  conductor. 

Experiment  also  shows  that  electric  charges  are  induced  on  a 
dielectric  which  is  placed  in  the  vicinity  of  a  charged  body,  in 
the  same  manner  that  charges  are  induced  on  a  conductor.     In 
the  case  of  a  solid  or  liquid  dielectric  separated  from  the  charged 
body  by  air  the  portion  of  the  dielectric  nearer  the  charged  body 
shows  a  charge  of  the  opposite  sign  to  that  on  the  charged  body, 
while  the  portion  of  the  dielectric  more  remote  from  the  charged 
body  shows  a  charge  of  the  same  sign  as  that  on  the  charged  body. 
However,  when  a  dielectric  which  is  thus  charged  by  induction 
is  separated  into  two  parts,  it  is  found  that  charges  are  produced 
on  the  walls  of  the  gap  between  the  two  parts,  a  negative  charge 
on  the  side  of  the  gap  nearer  the  positive  induced  charge  on  the 
original  surface  of  the  dielectric,  and  a  positive  charge  on  the  side 
of  the  gap  nearer  the  negative  charge  induced  on  the  original  sur- 
face of  the  dielectric.     Also,  when  the  dielectric  as  a  whole,  or 
any  portion  of  it,  is  removed  from  the  vicinity  of  the  charged 
body  which  induces  the  charges  on  it,  it  is  found  that  these  in- 
duced charges  entirely  disappear.     In  other  words,  the  phenome- 
non of  electrostatic  induction  in  a  dielectric  is  of  a  similar  nature 
to  the  phenomenon  of  magnetic  induction  in  a  magnetic  substance, 
except  that  it  is  impossible  to  produce  by  electrostatic  induction 
a   "  permanently    electrified "    dielectric.     The   above  facts   are 
true  only  of   a   dielectric   which  is   an  absolute  non-conductor. 
Every  dielectric,  however,  is  a  conductor  to  a  certain  extent,  and 
the  resultant  effect  produced  in  a  dielectric  when  it  is  placed 
in  the  vicinity  of  a  charged  body  is  therefore  a  combination  of 
the  effect  due  to  its  true  dielectric  or  insulating  property  and 
the  effect  due  to  its  conducting  property.     We  shall  see  later 
how  both  these  effects  may  be  taken  into  account. 

An  electric  charge  can  be  induced  only  at  the  surface  of 
separation  of  two  dissimilar  substances.  There  is  no  way  of 
determining  experimentally  "  on "  which  of  the  substances  in 
contact  the  charge  is  induced;  for  example,  when  a  piece  of 
glass  is  placed  in  air  in  the  vicinity  of  a  charged  body,  there 
is  no  way  of  determining  whether  the  induced  charge  at  the 
surface  of  separation  between  the  air  and  the  glass  is  "  on  " 


242  ELECTRICAL  ENGINEERING 

the  surface  of  the  air  or  "  on  "  the  surface  of  the  glass.  How- 
ever, just  as  in  the  discussion  of  the  phenomena  of  magnetism 
it  was  found  convenient  to  assume  that  air  was  non-magnetic  (and 
therefore  at  the  surface  of  separation  between  air  and  any  other 
substance  the  induced  poles  are  entirely  on  the  other  substance), 
so  in  the  discussion  of  the  phenomena  of  electrostatics,  it  is  found 
convenient  to  assume  that  at  the  surface  of  separation  between 
air*  and  any  other  substance,  there  is  no  charge  induced  on  the 
air.  In  general,  at  the  surface  of  separation  between  any  other 
two  substances,  whether  they  be  dielectrics  or  conductors,  it  will 
then  be  necessary,  in  order  to  account  for  the  observed  phenomena, 
to  assume  that  a  charge  is  induced  on  the  surface  of  both  these 
substances. 

130.  Point-Charges.  —  It  is  found  to  be  impossible  to  produce 
a  finite  electric  charge  at  a  point  in  space,  but  just  as  in  the  dis- 
cussion of  magnetic  phenomena  it  was  found  convenient  to  make 
use  of  the  conception  of  a  point-pole,  so  in  the  discussion  of 
electrostatic  phenomena  it  is   convenient  to   consider  a   charge 
of  finite  amount  concentrated  in  a  point.     Such  a  charge  may 
be  called  a  point-charge.     A  physical  approximation  to  a  point- 
charge  is  the  charge  on  a  small  area. 

131.  Properties  of  Electric  Charges.  —  When  all  the  electric 
charges  produced  in  any  manner  whatever,  including  the  charges 
induced  on  dielectrics,  are  taken  into  account,  it  is  found  that  the 
forces  produced  on  one  another  by  any  number  of  charged  bodies 
may  be  accounted  for  by  attributing  to  these  electric  charges  the 
following  properties: 

1.  Like  charges  repel  each  other  and  unlike  charges  attract 
each  other. 

2.  When  a  charge  of  one  sign  is  produced  on  any  body  an 
equal  and  opposite  charge  is  produced  either  on  the  same  or  on 
some  other  body. 

3.  Two  point-charges  q  and  qf  located  at  a  distance  r  apart 
repel  each  other  with  a  force  proportional  to  the  products  of  the 
quantities  q  and  q'  of  these  charges,  and  inversely  proportional 
to  the  square  of  the  distance  between  them,  independent  of  the 
nature  of  the  medium  between  them;  that  is,  with  a  force 

*Strictly,  this  assumption  is  permissible  only  for  one  definite  pressure; 
a  pressure  of  760  mm.  of  mercury  is  the  adopted  standard. 


ELECTROSTATICS  243 


where  ft  is  a  constant  depending  upon  the  units  in  which  q,  c[  ,  r 
and  /  are  measured.  Since  we  have  not  as  yet  specified  the  unit 
in  which  an  electric  charge  is  to  be  measured,  we  may  select  this 
unit  such  that  the  constant  k  in  the  above  equation  is  unity.  We 
then  have  that  the  force  of  repulsion  between  two  point-charges 
having  the  values  q  and  <?'  is 


r2 

When  q  and  (£  are  of  the  same  sign  this  force  is  positive,  and 
therefore  there  is  an  actual  repulsion  ;  when  q  and  of  are  of  opposite 
signs  this  force  is  negative,  that  is,  there  is  an  actual  attraction. 
This  agrees  with  the  first  property  stated  above.  The  line  of 
action  of  the  force  produced  by  one  point-charge  on  another  is 
the  line  drawn  between  them. 

The  resultant  force  produced  on  a  given  point-charge  qf  by 
any  number  of  point-charges  is  the  vector  sum  of  all  the  in- 
dividual forces  acting  on  that  charge,  that  is,  is 

*-<t  2^  (U) 

where  the  summation  £  —  includes  all  the  charges  in  the  vicinity, 

induced  or  otherwise,  on  conductors  or  dielectrics.  Since  the 
quantity  of  charge  induced  on  a  dielectric  depends  upon  the 
nature  of  the  dielectric,  this  resultant  force  will  also  depend  upon 
the  nature  of  the  surrounding  dielectrics,  but  the  force  due  di- 
rectly to  any  given  charge  q,  whether  on  a  conductor  or  induced 
on  a  dielectric,  is  independent  of  the  nature  of  the  medium  . 
between  q  and  g'. 

The  unit  of  electric  charge  is  defined  by  equation  (la);  that 
is,  a  unit  point-charge  is  a  charge  which  repels  with  a  force  of 
one  dyne  an  equal  point-charge  placed  one  centimeter  away.  This 
unit  is  called  the  c.  g.  s.  electrostatic  unit  of  charge. 

Since  the  properties  of  electric  charges  are  of  exactly  the  same 
form  as  the  properties  of  magnetic  poles,  with  the  one  exception 
that  a  charge  of  one  sign  can  exist  by  itself  on  a  conductor,  it 
follows  that  the  definitions  and  deductions  in  regard  to  the  proper- 
ties of  magnetic  poles  can  be  applied  directly  to  electric  charges, 
except  such  properties  as  were  deduced  as  a  consequence  of  the 


244  ELECTRICAL  ENGINEERING 

fact  that  equal  and  opposite  poles   always   exist   on  the  same 
body.* 

132.  Electrostatic  Field  of  Force.  —  Electrostatic  Intensity.  — 
Any  region  of  space  in  which  an  electric  charge  would  be  acted 
upon  by  a  mechanical  force  is  called  an  electrostatic  -field  of  force. 
The  intensity  of  the  electrostatic  field  at  any  point  is  denned  as 
the  force  in  dynes  which  would  be  exerted  by  the  agents  pro- 
ducing the  field  on  a  unit  point-charge  placed  at  that  point. 
The  names  electrostatic  intensity  and  electrostatic  force  are  also 
used  for  this  quantity.  The  electrostatic  intensity  at  any  point 
due  to  a  point-charge  q  at  a  distance  r  centimeters  away  is  then 


and  the  total  field  intensity  at  any  point  P  due  to  any  number 
of  point-charges  is  the  vector  sum 

TT     S9  (2fl) 

H=*S 

where  q  is  the  value  of  the  point-charge  at  any  point  and  r  is  the 
distance  in  centimeters  of  this  charge  from  the  point  P,  and  the 
summation  includes  all  the  charges,  induced  or  otherwise,  on  all 
the  conductors  and  dielectrics  in  the  field. 

The  total  field  intensity  at  any  point  P  due  to  any  charged 
surface  is  the  vector  integral 

(26) 


where  ds  is  any  elementary  area  of  the  surface,  <r  is  the  charge 
per  unit  area,  or  the  surface  destiny  of  the  charge  at  ds,  and  r  is 
the  distance  of  ds  from  P. 

133.  Lines  of  Electrostatic  Force.  —  Flux  of  Electrostatic 
Force.  —  Lines  of  electrostatic  force  may  be  drawn  in  the  same 
manner  as  lines  of  magnetic  force,  that  is,  lines  of  electrostatic 
force  are  lines  drawn  in  such  a  manner  that  the  direction  of  each 
line  at  each  point  coincides  with  the  direction  of  the  electrostatic 
intensity  at  that  point  and  the  number  of  these  lines  per  unit 
area  at  any  point,  normal  to  their  direction,  is  equal  to  the  value 
of  the  electrostatic  intensity  at  that  point.  The  number  of 

*The  same  symbols  will  as  a  rule  be  used  throughout  for  the  corresponding 
quantities;  in  any  problem  in  which  both  electrostatic  and  magnetic  quantities 
must  be  used,  the  former  may  be  distinguished  by  the  subscript  "e." 


ELECTROSTATICS  245 

these  lines  of  electrostatic  force  crossing  any  area  is  denned  as 
the  flux  of  electrostatic  force  across  that  area.  The  mathematical 
expression  for  the  flux  of  electrostatic  force  across  any  surface 
S  is  then 

$  =  f   (Hcosa)ds  (3) 

where  ds  is  any  elementary  area  of  this  surface,  H  the  elec- 
trostatic intensity  at  ds  and  a  the  angle  between  the  direction 
of  this  electrostatic  intensity  and  the  normal  to  ds.  Gauss's 
Theorem,  which  is  simply  a  consequence  of  the  inverse  square 
law,  also  holds  for  electric  charges;  that  is,  the  number  of  lines 
of  electrostatic  force  outward  across  any  closed  surface  is  equal 
to  the  algebraic  sum  of  the  charges  inside  the  surface.  The 
mathematical  expression  of  this  fact  is 


|(  H  cos  a  )  ds  =4  TT  2  q  (3a) 

where   f     represents  the  integral  over  the  closed  surface. 

134.  Lines  of  Electrisation.  —  We  have  already  seen  that 
when  a  charged  conductor  is  separated  into  two  parts  by  a  narrow 
gap,  there  is  no  electric  charge  produced  on  the  walls  of  this  gap. 
It  is  also  found  that  when  a  closed  cavity  of  any  kind  is  formed 
inside  a  charged  conductor,  no  electric  charge  is  produced  on  the 
walls  of  this  cavity  no  matter  how  the  external  surface  of  the 
conductor  is  charged.  A  charge  can  be  produced  on  the  walls 
of  such  a  cavity  only  by  introducing  a  charged  body  into  -this 
cavity.  Hence  in  a  charged  conductor  there  is  nothing  analogous 
to  lines  of  magnetisation  inside  a  magnetised  body. 

In  the  case  of  a  dielectric  charged  by  induction,  however,  we 
have  seen  that  in  general  charges  do  appear  on  the  walls  of  a 
gap  cut  in  the  dielectric,  just  as  magnetic  poles  appear  upon  the 
walls  of  a  gap  cut  in  a  magnetised  body.  It  is  possible,  however, 
just  as  in  the  case  of  a  magnetised  body,  to  cut  in  a  dielectric 
which  is  charged  by  induction  a  gap  in  such  a  direction  that 
no  charge  will  appear  on  the  walls  of  the  gap.  Hence  we  may 
consider  a  dielectric  which  is  charged  by  induction  to  be  made 
up  of  filaments  the  walls  of  each  of  which  have  such  a  direction 
that  were  this  filament  separated  from  the  rest  of  the  dielectric  by 
a  gap  of  infinitesimal  width,  there  would  be  no  charges  induced 
on  the  lateral  walls  of  this  filament.  The  two  ends  of  such  a 
filament  in  the  original  surface  of  the  dielectric  will  then  have 


246  ELECTRICAL  ENGINEERING 

equal  and  opposite  charges.     We  may  then  take  each  filament  of 
such  a  size  that  it  terminates  at  each  end  in  a  charge  which  has 

the   numerical    value  — .     The   positive    sense  of  such  a  fila- 
4?r 

ment  is  taken  as  the  direction  along  it  from  its  negative  to  its 
positive  end;  that  is,  these  filaments  are  considered  as  running 
through  the  dielectric  from  its  negatively  charged  end  to  its 
positively  charged  end.  These  filaments  are  called  lines  of 
electrisation.  The  intensity  of  electrisation  at  any  point  in  a 
body  is  defined  as  the  charge  per  unit,  area  which  would  appear 
on  a  gap  cut  in  the  body  at  this  point  perpendicular  to  the  line 
of  electrisation  through  this  point.  The  direction  of  the  intensity 
of  electrisation  at  any  point  is  taken  as  the  direction  of  the  line 
of  electrisation  through  this  point.  Lines  of  electrisation,  lines 
of  electrostatic  force,  and  the  electric  charge  induced  on  the  surface 
of  a  dielectric  are  then  related  to  one  another  in  exactly  the  same 
manner  as  lines  of  magnetisation,  lines  of  magnetic  force  and  the 
magnetic  poles  induced  on  the  surface  of  a  magnetic  body.  That 
is,  the  number  of  lines  of  force  ending  on  a  negative  charge  in- 
duced on  a  dielectric  is  equal  to  the  number  of  lines  of  electrisation 
originating  from  that  charge,  and  the  number  of  lines  of  electrisa- 
tion ending  on  a  positive  charge  induced  on  the  surface  of  a  dielec- 
tric is  equal  to  the  number  of  lines  of  electrostatic  force  originating 
from  that  charge.  The  relation  between  the  intensity  of  electrisa- 
tion at  any  point  in  the  surface  of  a  dielectric  and  the  charge 
induced  on  that  surface  is  then 

cr  =  J  cos  a  (4) 

where  cr  is  the  surface  density  of  this  induced  charge  at  this 
point,  J  the  intensity  of  electrisation  of  the  dielectric  at  this 
point,  and  a  the  angle  between  the  direction  of  the  intensity  of 
electrisation  at  this  point  and  the  direction  of  the  normal  drawn 
outward  from  the  surface  of  the  dielectric  at  this  point.  (See 
Article  46.) 

The  number  of  lines  of  electrisation  crossing  an  elementary 
surface  ds  the  normal  to  which  makes  an  angle  a  with  the 
direction  of  the  intensity  of  electrisation  is 

dN  =4  TT  (/  cos  a)  ds  (5) 

Since  we  have  assumed  (Article  129)  that  no  charges  are  in- 
duced on  the  surface  of  the  air  in  contact  with  any  other  sub- 
stance, the  intensity  of  electrisation  in  air  is  zero. 


ELECTROSTATICS  247 

It  should  be  clearly  kept  in  mind  that  the  above  relations 
hold  only  for  charges  induced  on  dielectrics.  Since  every  dielec- 
tric (other  than  a  perfect  vacuum)  is  a  conductor  to  a  slight 
extent,  charges  are  also  induced  on  the  dielectric  due  to  its  con- 
ducting property;  these  latter  charges,  however,  are  usually 
negligible  in  practical  work. 

135.  Lines  of  Electrostatic  Induction.  —  Flux  of  Electrostatic 
Induction.  —  The  algebraic  sum  of  the  number  of  lines  of  elec- 
trisation and  the  number  of  lines  of  electrostatic  force  across 
any  surface  in  an  electrostatic  field  is  called  the  number  of 
lines  of  electrostatic  induction  across  that  surface,  or  the  flux  of 
electrostatic  induction  across  that  surface.  Experiment  shows 
that,  except  in  the  special  case  of  certain  crystals,  the  intensity 
of  electrisation  at  any  point  in  a  dielectric  is  in  the  same  direction 
as  the  intensity  of  the  electrostatic  field.  Hence  the  mathemati- 
cal expression  for  the  number  of  lines  of  electrostatic  induction 
crossing  an  elementary  surface  dsn  normal  to  the  direction  of  the 
field  intensity  is 

J  ds 


where  H  is  the  intensity  of  the  electrostatic  field  at  ds,  and  J 
the  intensity  of  electrisation  in  the  dielectric  at  ds.  The  number 
of  lines  of  electrostatic  induction  per  unit  area  at  any  point  normal 
to  their  direction  is  called  the  electrostatic  flux  density*  at  this 
point.  The  relation  between  electrostatic  flux  density,  field  inten- 
sity and  intensity  of  electrisation  is  then 

B=H  +  4>rrJ  (6) 

Hence  the  flux  of  electrostatic  induction  across  any  elementary 
surface  ds  may  also  be  written 

d<f)=(B  cos  a)  ds  (7) 

where  B  is  the  electrostatic  flux  density  at  ds  and  a  is  the  angle 
between  the  direction  of  this  electrostatic  flux  density  and  the 
normal  to  ds. 

Since  the  intensity  of  electrisation  in  air  is  zero,  the  lines  of 
electrostatic  force  and  the  lines  of  electrostatic  induction  in  air 

*The  electrostatic  flux  density  divided  by  the  factor  4  TT  is  called  the  elec- 
trostatic polarisation  or  the  electric  displacement.     The  factor  4  IT  comes  in 
from  the  fact  that  one  line  of  polarisation  or  one  line  of  electric  displacemen 
is  considered  as  originating  from  each  unit  charge  on  a  conductor,  while  from 
a  unit  charge  on  a  conductor  there  originate  4  TT  lines  of  electrostatic  induction. 


248  ELECTRICAL  ENGINEERING 

are  identical,  and  therefore  the  electrostatic  intensity  and  the 
electrostatic  flux  density  in  air  are  equal. 

136.  Dielectric  Constant.  —  Experiment  shows  that  the  elec- 
trostatic flux  density  produced  in  any  dielectric  when  placed 
in  an  electrostatic  field  is  directly  proportional  to  the  resultant 
field  intensity,  and  except  in  the  special  case  of  certain  kinds  of 
crystals,  the  direction  of  the  electrostatic  flux  density  coincides 
with  the  direction  of  the  field  intensity.  The  ratio  of  the 
electrostatic  flux  density  to  the  field  intensity  is  called  the 
dielectric  constant  of  the  dielectric.  That  is,  calling  B  the  elec- 
trostatic flux  density  at  any  point  in  the  dielectric,  and  H  the 
resultant  electrostatic  intensity  at  this  point,  the  dielectric  con- 
stant of  the  dielectric  at  this  point  is 

K-*  (3) 

H 

The  dielectric  constant  is  exactly  analogous  to  magnetic  perme- 
ability. The  values  of  the  dielectric  constants  for  a  few  sub- 
stances are 

Solid  paraffin  2.29 

Paraffin  oil  1.92 

Ebonite  3.15 

Mica  6.64 

Glass  6.5  to  7.5 

Distilled  water  76 

Alcohol  26 

Ordinary  gases  1.000 

Perfect  vacuum  0.9995 

Combining  equations  (6)  and  (8)  we  have  that  H  +  47rJ  = 
KH  and  therefore  that 

J=(K-l)   H  (9) 

4?r 

Hence  in  any  dielectric  for  which  the  dielectric  constant  is  unity 
the  intensity  of  electrisation  is  zero,  and  therefore  there  are  no 
lines  of  electrisation,  and  hence  there  are  no  charges  induced  on 
the  surface  of  such  a  dielectric.  In  a  dielectric  for  which  the 
dielectric  constant  is  greater  than  unity,  the  intensity  of  electri- 
sation is  not  zero  but  a  positive  quantity,  and  therefore  the  field 
intensity  is  less  than  the  flux  density ;  hence  the  number  of  lines 
of  electrostatic  force  inside  a  dielectric  of  which  the  dielectric 


ELECTROSTATICS  249 

constant  is  greater  than  unity  is  always  less  than  the  number 
of  lines  of  electrostatic  induction. 

137.  A  Closed  Hollow  Conductor  an  Electrostatic   Screen.  - 
It  is  found  by  experiment  that  an  electric  charge  cannot  be  pro- 
duced on  the   inside  surface  of   a  hollow   closed  conductor  by 
charging  the  conductor  either  by  contact  with,  or  by  induction 
from,   any   agent   whatever  outside  the   conductor;  nor  can   an 
electrostatic  field  be  produced  inside  the  space  enclosed  by  the 
conductor   by   any   agent   whatever   exterior  to    the    conductor, 
provided  in  each  case  there  is  no  electric  current  in  the  conductor. 
These  two  important  facts  were  discovered  by  Faraday,  who 
built  a  large  insulated  metal  chamber  in  which  he  set  up  the 
most  delicate  measuring  instruments  he  could  devise;  he  was 
unable   to    detect    the   slightest,  electrostatic   effect   inside   this 
chamber  no  matter  how  highly  the  chamber  was  electrified  on 
the  outside.     All   deductions  from  this   discovery  of   Faraday's 
have  been  found  to  be  in  accord  with  experiment,  and  we  there- 
fore accept  as  a  fundamental  law  of  electrostatics  that  any  region 
completely  enclosed  by  a  conductor  is  absolutely  protected  from 
external  electrostatic  effects.     It  is  also  found  that  a  metal  gauze 
or  cage  forms  a  practically  perfect  screen. 

138.  The  Resultant  Electrostatic  Intensity  Within  the  Substance 
of  a  Conductor  in  which  there  is  no  Electric  Current  is  Always  Zero. 

—  An  important  deduction  from  Faraday's  discovery  is  that 
the  resultant  electrostatic  intensity  within  the  substance  of  a 
charged  conductor  is  zero,  provided  there  is  no  current  in  the  con- 
ductor, no  matter  how  the  conductor  may  be  charged.  For,  should 
a  closed  cavity  be  made  in  this  conductor,  no  charge  would  appear 
on  the  walls  of  this  cavity,  and  therefore  the  original  electro- 
static field  would  not  be  altered;  therefore  the  electrostatic  field 
inside  this  cavity  must  be  the  same  as  originally  existed  hi  the 
conducting  material  which  filled  this  cavity,  since  the  electro- 
static intensity  at  any  point  due  to  any  given  distribution  of 
electric  charges  is  independent  *of  the  material  at  the  point  in 
question.  But  the  electrostatic  intensity  in  the  cavity  is  zero, 
and  since  the  charges  producing  the  field  are  not  altered  by 
forming  the  cavity,  the  electrostatic  intensity  in  the  conducting 
material  which  originally  filled  this  cavity  must  also  have  been 
zero. 

Since  the  resultant  intensity  in  a  conductor  is  zero,  it  follows 


250  ELECTRICAL  ENGINEERING 

that  when  a  conductor  is  placed  in  an  electrostatic  field,  the  induced 
charges  which  appear  must  be  distributed  in  such  a  manner  that 
the  field  within  the  substance  of  the  conductor  due  to  these  in- 
duced charges  is  just  equal  and  opposite  to  the  original  field  in 
the  space  occupied  by  the  conductor. 

139.  The  Total  Charge  Within  any  Region  Completely  En- 
closed by  a  Conductor  is  Always  Zero.  —  We  have  just  seen  that 
the  electrostatic  intensity  is  zero  within  the  substance  of  a  con- 
ductor in  which  there  is  no  electric  current.  Hence  when  any 
region  of  space  is  completely  enclosed  by  such  a  conductor  the 
total  flux  of  electrostatic  intensity  out  through  any  surface 
drawn  within  the  substance  of  this  conducting  shell  is  zero,  since 
there  are  no  lines  of  electrostatic  force  crossing  this  surface. 
Hence,  by  Gauss's  Theorem,  the  total  charge  inside  such  a  surface 
is  zero.  Also,  since  it  is  one  of  the  fundamental  properties  of 
electric  charges  that  a  charge  of  one  sign  cannot  be  produced 
without  at  the  same  time  producing  an  equal  charge  of  the  op- 
posite sign,  the  total  charge  outside  such  a  closed  surface  must 
also  be  zero.  Hence  a  closed  conducting  surface  divides  all 
space  into  two  parts  in,  each  of  which  parts  the  total  charge  is 
zero.  For  example,  let  A  be  an  insulated  conductor  which  has 


-Q 


Fig.  74. 

a  charge  of  Q  units ;  then  on  some  other  conductor  or  conductors  B 
there  must  be  an  equal  and  opposite  charge  of  —  Q  units.  Let  A 
be  completely  surrounded  by  an  insulated  conducting  shell  C; 
there  must  then  be  induced  on  the  inside  of  this  shell  a  charge 
of  —  Q  units  and  on  the  outside  of  this  shell  a  charge  of  +  Q  units. 
140.  The  Electrostatic  Intensity  Just  Outside  a  Charged  Con- 
ductor is  Normal  to  the  Surface  of  the  Conductor.  — By  the  same 
process  of  reasoning  as  that  employed  in  Article  50,  by  which 
it  was  shown  that  the  tangential  components  of  the  magnetic 
field  intensities  just  inside  and  just  outside  a  magnetically  charged 


ELECTROSTATICS  251 

surface  at  any  point  are  equal,  it  can  be  shown  that  the  tangential 
components  of  the  electrostatic  intensities  just  inside  and  just 
outside  an  electrically  charged  surface  are  also  equal  at  each  point 
of  the  surface.  In  the  case  of  a  charged  conductor  in  which  there 
is  no  electric  current  we  have  just  seen  that  the  resultant  electro- 
static intensity  inside  the  surface  is  zero,  and  therefore  inside  the 
surface  the  tangential  component  is  zero.  Hence  the  tangential 
component  of  electrostatic  intensity  just  outside  the  surface  must 
also  be  zero.  (When  there  is  an  electric  current  in  the  conductor 
the  tangential  component  of  the  electrostatic  intensity  at  the 

surface  of  the  conductor  is  not  zero,  but  is  equal  to  — £ where 

3X1010 

p  is  the  specific  resistance  of  the  conductor  in  ohms  per  centimeter 
cube  and  cr  is  the  current  density  in  abamperes  per  square  centi- 
meter, see  Articles  102  and  148;  this  quantity,  however,  is  usually 
negligibly  small.)  Hence  the  important  result:  the  resultant  elec- 
trostatic intensity  just  outside  the  surface  of  a  charged  conductor 
is  normal  to  the  surface  of  the  conductor;  or,  stated  in  other  words, 
the  resultant  lines  of  electrostatic  force  are  always  perpendicular 
to  the  surface  of  a  conductor  where  they  enter  or  leave  it.  This 
relation  is  absolutely  true  in  case  there  is  no  electric  current  in 
the  conductor  and  also  holds  to  a  close  degree  of  approximation 
in  any  practical  case  when  there  is  a  current  in  the  conductor. 
141.  The  Electrostatic  Flux  Density  Just  Outside  a  Charged 
Conductor  is  Independent  of  the  Nature  of  the  Surrounding  Di- 
electric. —  Let  the  dielectric  in  contact  with  the  conductor  have 
a  dielectric  constant  K,  and  let  the  surface  density  of  the  charge 
on  the  conductor  at  any  point  P  be  cr.  A  charge  will  also  be 
induced  on  the  surface  of  the  dielectric  in  contact  with  the  con- 
ductor at  P;  let  cr'  be  the  value  of  this  induced  charge.  Draw 
around  P  a  small  right  cylinder  with 
its  axis  normal  to  the  surface  of  the 
conductor  at  P  and  one  end  of  the 
cylinder  inside  the  surface  and  the 
other  outside  the  surface.  Let  H  be 
resultant  electrostatic  intensity  at  P 
just  outside  the  surface  of  the  con- 
ductor, and  let  the  end  of  the  cylinder 
have  the  area  ds.  Then  when  the  ends 
of  the  cylinder  are  infinitely  close  to  Fig.  75. 


252  ELECTRICAL  ENGINEERING 

the  surface,  the  electrostatic  intensity  outside  the  conductor 
will  be  normal  to  ds.  Hence  the  total  flux  of  electrostatic 
force  across  the  outside  end  of  the  cylinder  will  be  Hds. 
Across  the  lateral  walls  of  the  cylinder  outside  the  conducting 
surface  the  flux  of  force  will  be  zero,  since  these  lateral  walls 
are  parallel  to  the  direction  of  the  field  intensity.  Across  the 
walls  of  the  portion  of  the  cylinder  inside  the  surface  the  flux 
of  force  will  likewise  be  zero,  since  there  is  no  electrostatic  field 
inside  the  conductor.  Hence  the  total  flux  of  force  outward 
through  the  walls  of  this  cylinder  is  Hds.  By  Gauss's  Theorem 
this  must  be  equal  to  4  TT  times  the  total  charge  inside  this  cylinder, 
that  is,  equal  to  4  TT  (cr  +  <r')  ds.  Hence 

#=477(0-  +0-0 

But  since  the  lines  of  electrisation  (which  coincide  in  direction 
with  the  lines  of  force)  are  perpendicular  to  the  surface  at  P  and 
are  into  the  dielectric  from  its  surface,  the  intensity  of  electrisa- 
tion of  the  dielectric  just  outside  the  conductor  at  P  is 

J=-cr' 

see  equation  (4).  From  equation  (6)  the  electrostatic  flux  density 
at  P  is  B  =  H  +  4:7rJ.  Therefore  substituting  for  H  and  /  their 
values  in  terms  of  cr  and  cr',  we  have 


That  is,  the  electrostatic  flux  density  at  any  point  just  outside  a 
conductor  is 

5=4770-  (10) 

where  cr  is  the  surface  density  of  the  charge  on  the  conductor  at 
this  point,  and  is  independent  of  the  nature  of  the  dielectric  in  con- 
tact with  the  conductor  at  this  point.*  The  resultant  electrostatic 
intensity  just  outside  a  charged  conductor,  on  the  other  hand, 
does  depend  upon  the  nature  of  the  dielectric  in  contact  with  the 
conductor,  and  is  equal  to 

tf=l^  (11) 

K 

where  K  is  the  dielectric  constant  of  the  dielectric  in  contact 
with  the  conductor  at  the  point  in  question. 

An  important  difference  between  lines  of  magnetic  induction 

*In  the  deduction  of  this  relation  the  contact  charge,  if  any,  produced  on  the 
surface  of,  the  dielectric,  is  neglected.  In  case  there  is  a  contact  charge  on  the 
dielectric  this  same  relation  holds  when  O"  is  taken  to  represent  the  surface 
density  of  .the  charge  on  the  conductor  plus  the  contact  charge  on  the  dielectric. 


ELECTROSTATICS  253 

and  lines  of  electrostatic  induction  should  be  noted  in  this  con- 
nection. Lines  of  magnetic  induction  are  always  closed  curves. 
On  the  other  hand,  lines  of  electrostatic  induction  which  are 
due  entirely  to  electric  charges  originate  from  and  end  on  con- 
ductors (or  contact  charges  on  dielectrics) ;  4  TT  of  them  originate 
from  each  unit  positive  charge  on  a  conductor  and  4  TT  of  them 
end  on  each  unit  negative  charge  on  a  conductor.  They  do  not, 
however,  end  on  charges  induced  on  dielectrics,  but  pass  through 
these  charges  (see  Article  54),  in  general  making  an  abrupt  change 
in  direction  at  the  surface  of  a  dielectric  on  which  a  charge 
is  induced.  Lines  of  electrostatic  force,  however,  are  strictly 
analogous  to  lines  of  magnetic  force ;  4  77  of  them  originate  from 
every  unit  positive  charge  and  4  TT  of  them  end  on  every  unit 
negative  charge,  whether  these  charges  be  on  a  conductor  or  are 
induced  charges  on  a  dielectric.  Hence  in  any  dielectric  which 
has  a  dielectric  constant  greater  than  unity  there  are  fewer  lines 
of  force  than  lines  of  induction,  for  the  lines  of  induction  all  pass 
through  the  dielectric  while  some  of  the  lines  of  force  end  on  its 
surface. 

142.  Conditions  which  Must  be  Satisfied  at  any  Surface  in  an 
Electrostatic  Field.  —  By  identically  the  same  process  of  reasoning 
as  that  employed  in  Articles  49  and  50  it  can  be  shown  that  at 
every  surface  in  an  electrostatic  field  other  than  a  conducting 
surface  the  following  conditions  must  be  satisfied: 

1.  The  normal  components  of  the  electrostatic  induction  on 
the  two  sides  of  any  surface  at  any  given  point  in  this  surface 
must  be  equal. 

2.  The  tangential  components  of  the  electrostatic  intensity 
on  the  two  sides  of  any  surface  at  any  given  point  in  this  surface 
must  be  equal. 

At  the  surface  of  a  conductor  the  surface  conditions  are,  as 
we  have  just  seen,  that 

1.  Both  the  electrostatic  intensity  and  the  electrostatic  induc- 
tion inside  the  surface  are  zero. 

2.  Both  the  electrostatic  intensity  and  the  electrostatic  induc- 
tion just  outside  the  surface  are  normal  to  the  surface. 

From  these  surface  conditions  the  distribution  of  the  charges 
on  conductors  and  those  induced  on  dielectrics  may  be  calculated 
in  certain  simple  cases. 

143.  Electrostatic  Potential.  —  The  work  that  would  be  done 


254  ELECTRICAL   ENGINEERING 

by  the  agents  producing  an  electrostatic  field  on  a  unit  positive 
point-charge,  were  such  a  point-charge  moved  from  any  point 
P  in  the  field  to  infinity,  is  called  the  electrostatic  potential  of  the 
field  at  the  point  P.  The  electrostatic  potential  at  any  point  P 
due  to  a  point-charge  q  at  a  distance  r  from  P  is  then 

VJL  (12) 

r 

and  is  independent  of  the  path  over  which  the  charge  is  moved 
from  P  to  infinity,  and  therefore  depends  only  upon  the  relative 
position  of  the  charge  and  the  point  P.  (See  Article  59.)  Since 
electrostatic  potential  is  the  ratio  of  work  to  charge,  and  since 
both  work  and  charge  are  scalar  quantities,  electrostatic  poten- 
tial is  also  a  scalar  quantity.  Hence  the  resultant  electrostatic 
potential  at  any  point  due  to  any  number  of  electric  charges  is 
the  algebraic  sum  of  the  potentials  due  to  each  charge  separately. 
Therefore  the  electrostatic  potential  at  any  point  P  due  to  a 
charged  surface  of  any  kind  is 

"  o-ds  (12o) 


•L 


8 

where  ds  represents  any  elementary  area  of  the  surface,  cr  the 
surface  density  of  the  charge  at  ds,  and  r  the  distance  of  the  point 
P  from  ds. 

144.  Difference  of  Electrostatic  Potential.  —  The  difference  of 
electrostatic  potential  between  any  two  points  1  and  2,  or  specifi- 
cally, the  drop  in  electrostatic  potential  from  the  point  1  to  the 
point  2,  is  the  work  that  would  be  done  by  the  agents  producing 
the  electrostatic  field  on  a  unit  positive  point-charge  were  such 
a  charge  moved  from  the  point  1  to  the  point  2.  Calling  dl 
an  elementary  length  in  any  path  from  the  point  1  to  the  point 
2,  H  the  electrostatic  intensity  at  dl,  and  0  the  angle  between 
the  direction  of  the  electrostatic  intensity  H  and  the  direction 
of  the  length  dl,  the  drop  of  electrostatic  potential  from  1  to  2 
along  this  path  is  then 

/*2 

(HcosO)dl  (13) 

When  there  is  no  contact  or  induced  electromotive  force  in  the 
path  from  1  to  2,  the  drop  of  electrostatic  potential  from  any  point 
1  to  any  point  2  is  independent  of  the  path  over  which  the  charge 


ELECTROSTATICS  255 

is  moved,  and  therefore  under  this  condition  the  drop  of  electro- 
static potential  around  any  closed  path  in  the  field  is  zero. 

However,  as  we  shall  see  presently,  when  there  is  any  contact 
or  induced  electromotive  force  in  the  field,  this  contact  or  induced 
electromotive  force  also  produces  an  electrostatic  field  the  lines 
of  force  representing  which  are  closed  loops  passing  through  the 
surface  of  contact  in  the  case  of  a  contact  electromotive  force 
and  linking  the  lines  of  magnetic  induction  in  the  case  of  an  in- 
duced electromotive  force.  In  such  a  case  the  drop  of  electrostatic 
potential  around  any  closed  loop  is  equal  to  the  algebraic  sum  of 
the  contact  and  induced  electromotive  forces  in  the  loop,  provided 
the  drop  of  electrostatic  potential  and  the  electromotive  forces 
are  expressed  in  the  same  unit.  (See  Article  148.)  Hence,  when 
there  are  any  contact  or  induced  electromotive  forces  in  an  electro- 
static field  the  difference  of  electrostatic  potential  between  any 
two  points  depends  not  only  upon  the  position  of  the  two  points 
but  also  upon  the  path  over  which  the  unit  charge  is  moved.  For 
example,  in  the  conductor  forming  the  armature  winding  of  a 
continuous  current  generator  the  resultant  electrostatic  intensity 
is  zero  when  there  is  no  current  in  the  armature  conductors  (see 
Article  138).  Hence  the  drop  of  electrostatic  potential  through 
the  winding  from  the  positive  to  the  negative  terminal  is  zero,  but 
the  drop  of  electrostatic  potential  from  the  positive  to  the  negative 
terminal  through  the  surrounding  air  is  equal  to  the  electromotive 
force  of  the  generator.  Compare  also  with  the  relation  between 
drop  of  magnetic  potential  around  a  closed  loop  and  the  magneto- 
motive force  linked  by  this  loop,  Article  113. 

145.  Electrostatic  Equipotential  Surfaces.  —  An  electrostatic 
equipotential  surface  is  a  surface  drawn  in  an  electrostatic  field 
in  such  a  manner  that  the  drop  of  electrostatic  potential  along  any 
path  in  the  surface  is  zero.  Such  a  surface  will  intersect  the  lines 
of  electrostatic  force  at  right  angles.  For,  were  any  line  of  force 
not  perpendicular  to  the  surface  where  it  crosses  the  surface,  the 
electrostatic  intensity  at  this  point  of  intersection  would  have 
a  component  parallel  to  the  surface,  and  therefore  work  would 
be  done  on  a  unit1  point-charge  were  the  latter  moved  along  the 
surface  in  the  direction  of  this  component;  but  this  is  contrary 
to  the  condition  that  the  surface  is  drawn  in  such  a  manner  that 
there  is  no  difference  of  potential  between  any  two  points  in  its 
surface. 


256  ELECTRICAL  ENGINEERING 

When  a  conductor  in  which  there  is  no  electromotive  force 
is  placed  in  the  electrostatic  field,  the  conductor  must  become 
an  equipotential  surface,  for  we  have  already  seen  that  the  electro- 
static intensity  is  always  normal  to  the  surface  of  a  conductor  in 
which  there  is  no  electric  current,  and  therefore  the  drop  of 
potential  along  any  path  in  the  surface  of  the  conductor  is  zero. 

146.  Parallel  Plate  Electrometer.  —  The  following  exan  pie 
wTill  illustrate  the  principles  stated  above,  and  will  at  the  same 
time  show  how  electric  charge  and  difference  of  electrostatic 
potential  may  be  measured.  By  the  same  process  of  reasoning 
as  that  employed  in  Article  37  it  can  be  shown  that  the  electro- 
static intensity  due  to  a  uniformly  charged  plane  surface  is  in 
the  direction  normal  to  this  surface  at  any  point  at  a  perpendic- 
ular distance  from  the  surface  small  compared  with  the  distance 
of  the  point  from  the  perimeter  of  the  surface,  and  that  its  value  is 

#=27rcrn  (14) 

where  crn  is  the  surface  density  of  the  net  charge  at  this  surface. 
By  net  charge  is  meant  the  algebraic  sum  of  the  charge  on  the 
conductor,  the  contact  charge  (if  any)  on  the  dielectric  in  contact 
with  the  conductor,  and  the  charge  induced  on  the  dielectric. 
Consider  two  equal  plane  metal  discs  A  and  B  placed  parallel 
.  a'  and  opposite  to  each  other  and  im- 

mersed  in   a   dielectric  of  which  the 

B  • —f dielectric  constant  is   K,  and  let  this 

Fig.  76.  dielectric  completely  fill  all  the  sur- 

rounding space.  Let  the  plates  A  and  B  be  given  charges  of 
Q  and  —  Q  units  respectively.  The  only  charges  induced  on 
the  dielectric  will  then  be  those  at  the  surface  of  separation 
between  the  metal  plates  and  the  dielectric.  The  charges  Q 
and  —  Q  and  the  charges  induced  on  the  dielectric  must  be  dis- 
tributed in  such  a  manner  that  (1)  the  electrostatic  intensity  at 
any  point  inside  the  plates  is  zero,  and  (2)  the  electrostatic  in- 
tensity at  each  point  in  the  dielectric  infinitely  close  to  the  surface 
of  the. plates  is  normal  to  the  surface  at  this  point.  A  uniform 
distribution  of  the  net  charge  at  the  two  surfaces  a  and  b  which 
face  each  other  and  no  charge  on  the  "  back"  surfaces  a'  and  I/  will 
satisfy  these  conditions  for  all  points  except  those  near  the  edges 
of  the  plates.  For,  calling  crn  and— crn  the  surface  densities  of 
the  net  charge  at  the  surfaces  a  and  b  respectively,  the  resul- 
tant intensity  (from  equation  14)  inside  each  plate  will  be 


ELECTROSTATICS  257 

2  TT  crn—  2  TT  <rw  =  0,  and  the  intensity  at  any  point  in  the  space 
between  the  plates  will  be  2  TT  crn  +  2  TT  crn  =4  TT  crn  and  will  be 
perpendicular  to  these  surfaces,  while  at  each  point  infinitely  close 
to  the  surfaces  a'  and  b'  the  intensity  will  be  2  TT  crn—  2  TT  crn  =0, 
which  relations  satisfy  the  surface  conditions  1  and  2.  It  can  also 
be  shown  in  that  there  is  only  one  possible  distribution  of  charges 
which  can  satisfy  these  surface  conditions  ;  hence  at  all  points  at  a 
considerable  distance  from  the  edges  of  the  plates  the  charges  at 
the  surfaces  of  separation  between  the  dielectric  and  the  plates 
must  be  distributed  uniformly  over  the  surfaces  a  and  b  and  there 
can  be  no  charge  on  the  surfaces  of  and  bf  (except  near  their 
edges)  . 

Let  cr  be  the  value  of  the  total  charge  on  the  conductor  and 
the  contact  charge,  if  any,  on  the  dielectric  at  each  point  of  the 
surface  a  and  cr'  the  value  of  the  charge  induced  on  the  dielectric  at 
the  surface  a.  Then  crn  =cr  +  cr'  =cr—  /,  where  J  is  the  intensity 
of  electrisation  of  the  dielectric  in  the  direction  from  a  to  b.  But 

(  Jf      1  ^  J-T 

J  =-  -  -  —  ,  equation  (9a),  whence  the  resultant  field  intensity 

4  7T 

at  any  point  in  the  space  between  the  plates,  not  too  close  to  their 
edges,  is 


K 

where  cr  is  the  surface  density  of  the  charge  on  the  conductor.  (Com- 
pare with  equation  11.)  This  electrostatic  intensity  is  therefore 
perpendicular  to  the  surfaces  of  the  plates,  and  the  electro- 
static field  between  the  two  plates  is  a  uniform  field  except  for 
points  not  too  close  to  the  edges  of  the  plates. 

The  force  of  attraction  between  the  two  plates  may  be  readily 

calculated.     For,  of  the  resultant  intensity  H  =  —    —  ,   half  must 

K. 

be  due  to  each  plate,  that  is  the  resultant  field  intensity  due  to  each 

plate  is  -^-  —  .    This,  then,  is  the  value  of  the  force  due  to  the  plate 
K. 

A,  say,  on  each  unit  charge  on  the  plate  B  (or,  vice  versa,  the  force 
due  to  the  plate  B  on  each  unit  charge  on  the  plate  A).  Hence 

the  pull  on  each  unit  area  of  the  plate  A  is  —      —  ,  and  when  the 

K 

linear  dimensions  of  the  plates  are  equal  and  are  large  compared 


258  ELECTRICAL  ENGINEERING 

to  their  distance  apart,  so  that  the  non-uniformity  of  the  distribu- 
tion of  the  charge  at  their  edges  may  be  neglected,  the  total  force 
of  attraction  between  the  two  plates  is 


K  KS 

and  therefore 


where  S  is  the  area  of  each  plate  and  Q  the  total  charge  on  each 
plate.     The  force   F  may  be  readily  measured  by  suspending 
one  of  the  plates  from  one  arm  of  a  suitable  chemical  balance. 
When  the  plates  are  separated  by  air,  the  force  of  attraction 

2    TT   O2 

between  the  two  plates  is  F= ,  since  the  dielectric  con- 

S 

stant  of  air  is  unity.  In  this  case  the  charge  on  each  plate  is 
numerically  equal  to 

^SF 
-  (156) 

Hence,  by  measuring  the  force  of  attraction  between  the  two 
plates  the  charge  on  each  p  ate  may  be  calculated.  Again, 
from  the  fact  that  the  electrostatic  field  between  the  two  plates 
is  uniform,  the  difference  of  electrostatic  potential  between  the 
plates  may  also  be  calculated.  For,  calling  D  the  distance 
between  the  two  plates,  we  have,  from  equation  (13),  that  this 
difference  of  electrostatic  potential  is 

JD       _47ro-  D_4ir  Q  D_       |g  77  F 
/r<H--r--nnr-Ninr_      (16) 

When  the  plates  are  separated  by  air,  the  dielectric  constant  is 
unity,  and  therefore  the  difference  of  electrostatic  potential 
between  the  two  plates  is 

(16a) 


The  arrangement  of  parallel  plates  above  described,  which 
arrangement  is  called  a  parallel  plate  electrometer,  gives  us  a 
means  of  measuring  both  electrostatic  charge  and  electrostatic 
potential  difference.  The  above  calculations  are  all  based  upon 
the  assumption  that  the  non-uniformity  of  the  charge  near  the 
edges  of  the  plates  may  be  neglected.  The  condition  of  uni- 


ELECTROSTATICS  259 

formity  of  charge  on  the  suspended  plate  and  uniformity  of  the 
electrostatic  field  acting  on  it  may  be  more  closely  realized  by 
suspending  only  the  central  portion  of  the  upper  plate,  and  letting 
the  rest  of  this  plate  form  a  flat  metal  ring  which  is  kept  station- 
ary but  connected  by  a  conductor  to  the  suspended  portion. 
That  is,  the  upper  plate  is  made  in  two  portions;  a  central  disc 
suspended  from  the  arm  of  the  balance  and  an  outside  "  guard 
ring  "  with  its  lower  surface  flush  with  the  lower  surface  of  the 
ring  and  separated  from  the  disc  by  a  narrow  air  gap,  the  ring 
and  plate  being  conductively  connected  through  the  metal  wires 
from  which  the  disc  is  suspended.  The  surface  S  in  the  above 
formulas  is  then  the  area  of  the  surface  of  this  disc,  and  the  charge 
Q  is  the  charge  on  this  disc. 

147.  Relation  between  Electrostatic  Charge  and  Quantity  of 
Electricity.  —  Consider  an  electrometer  of  the  form  described  in 
the  preceding  article,  and  for  simplicity  let  the  plates  of  the 
electrometer  be  of  the  same  area  and  let  there  be  no  guard  ring. 
When  the  two  plates  are  connected  by  a  wire,  any  charge  which 

W 


B 


Fig.  77. 

might  have  been  on  them  disappears,  and  there  is  no  appreciable 
force  of  attraction  between  them.  The  plates  are  then  un- 
charged and  are  at  the  same  electrostatic  potential  and  also  at 
the  same  electric  potential,  as  defined  in  Article  87,  since  there  is 
no  current  in  the  wire  connecting  them.  (If  the  plates  are  orig- 
inally charged  before  they  are  connected  by  the  wire  a  momen- 
tary current  will  be  established  in  the  wire,  as  we  shall  see  pres- 
ently, but  this  current  lasts  only  for  a  small  fraction  of  a  second.) 
Arrange  two  insulated  wires  W  and  W  in  series  respectively 
with  the  ballistic  galvanometers  G  and  Gf ',  and  connect  one  end 
of  these  wires  respectively  to  the  positive  and  negative  terminals 
of  a  generator  or  other  source  of  electromotive  force.  Remove 


260  ELECTRICAL  ENGINEERING 

the  wire  connecting  the  two  plates  of  the  electrometer  and  con- 
nect the  other  ends  of  the  wires  W  and  W  to  the  plates  A  and  B 
respectively.  The  following  phenomena  will  be  observed  : 

1.  A  momentary  electric  current  is  established  in   the   two 
galvanometers,  as  shown  by  their  deflections,  and  therefore  a 
certain  quantity  of  electricity  as  defined  in  Article  80  is  trans- 
ferred across  each  section  of  the  wire  forming  the  galvanometer 
windings  and  the  wires  W  and  W.     If  the  galvanometers  have 
been  calibrated  (see  Article  109)  the  quantity  of  electricity  trans- 
ferred through  each  can  be  measured  and  the  direction  of  its 
transfer  can  be  observed.     It  will  be  found  that  equal  quantities 
of  electricity  are  transferred  through  each  galvanometer  (provided 
the  connecting  wires  are  small)  and  that  the  direction  of  transfer 
or  "  flow  "  of  electricity  is  from  the  plate  B  which  is  connected 
to  the  negative  terminal  of  the  generator  and  towards  the  plate 
A  which  is  connected  to  the  positive  terminal  of  the  generator. 

2.  The  plates  A  and  B  will  become  charged,  as  shown  by  their 
mutual  attraction,  the  plate  A  positively  and  the  plate  B  nega- 
tively.    The  charges  given  the  two  plates  will  also  be  numerically 
equal,  as  can  be  tested  by  observing  the  force  which  each  produces 
on  some  other  charged  conductor  when  placed  successively  in  the 
same  relative  position  with  respect  to  the  latter. 

When  the  value  of  the  impressed  electromotive  force,  or 
trie  distance  apart  of  the  plates,  or  the  size  of  the  plates,  or  the 
dielectric  between  them,  is  altered,  it  is  found  that  the  charge 
given  each  plate  also  changes ;  the  quantity  of  electricity  trans- 
ferred through  each  galvanometer  likewise  changes.  In  each 
case,  however,  it  is  found  that  the  charge  given  each  plate  is 
directly  proportional  to  the  quantity  of  electricity  transferred 
through  each  galvanometer.  When  the  charge  given  each  plate 
is  measured  in  electrostatic  units,  as  defined  in  Article  131,  and 
the  quantity  of  electricity  transferred  through  each  galvanometer 
is  measured  in  electromagnetic  units  as  defined  in  Article  80,  it  is 
found  that  the  numerical  value  of  the  charge  given  each  plate 
is  equal  (approximately)  to  3  x  1010  times  the  numerical  value 
of  the  quantity  of  electricity  transferred  through  each  galva- 
nometer. This  figure  3  x  1010  is  approximately  equal  to  the 
velocity  of  light  in  air.  Consequently,  if  as  the  unit  of  charge 
is  taken  a  unit  3  x  1010  times  the  size  of  the  electrostatic  unit 
as  defined  in  Article  131,  the  numerical  value  of  the  electric 


ELECTROSTATICS  261 

charge  given  each  plate  will  be  equal  to  the  number  of  electro- 
magnet units  of  quantity  of  electricity  transferred  to  it ;  similarly, 
if  as  the  unit  of  charge  is  taken  a  unit  3  x  109  times  the  size  of 
an  electrostatic  unit  the  numerical  value  of  the  electric  charge 
given  each  plate  will  be  equal  to  the  number  of  coulombs  trans- 
ferred to  it.  Electric  charge  and  quantity  of  electricity  may 
then  be  measured  in  the  same  units ;  these  units  are  related  as 
follows : 

1  c.  g.  s.  electromagnetic  unit  or  abcoulomb 

=3  x  1010  c.  g.  s.  electrostatic  units 

1  coulomb  =3  x  109  c.  g.  s.  electrostatic  units 

(The  figure  3.00  x  1010  is  about  as  accurate  as  can  be  determined 
by  a  direct  comparison  of  electromagnetic  and  electrostatic  units. 
The  electromagnetic  theory  of  light  requires  that  ratio  between 
the  electromagnetic  and  electrostatic  units  be  exactly  equal  to 
the  velocity  of  light  in  air,  and  the  velocity  of  light  by  direct 
optical  methods  is  found  to  be  2.998X  1010  centimeters  per  second. 
The  experimental  fact  that  the  ratio  of  the  two  kinds  of  units 
is,  as  closely  as  can  be  measured,  equal  to  the  velocity  of  light 
in  air,  is  strong  evidence  for  our  belief  in  the  truth  of  the  electro- 
magnetic theory.) 

The  phenomena  just  described  are  reversible;  that  is,  when 
the  charge  on  a  conductor  disappears  a  current  is  established  in 
the  conductor  which  transfers  from  each  element  of  the  surface 
of  the  conductor  a  quantity  of  electricity  equal  to  the  charge 
originally  on  this  surface.  For  example,  when  the  two  plates 
of  the  electrometer  described  above  are  disconnected  from  the 
generator  and  then  connected  to  each  other  through  a  ballistic 
galvanometer,  it  is  found  that  the  quantity  of  electricity  trans- 
ferred through  the  galvanometer  is  exactly  equal  to  the  quantity 
of  electricity  originally  transferred  through  the  galvanometers 
when  the  plates  were  charged  (provided  the  plates  are  otherwise 
perfectly  insulated),  and  the  transfer  of  charge  is  in  the  direction 
from  the  positive  to  the  negatively  charged  plate ;  the  plates  also 
no  longer  attract  each  other,  i.e.,  the  charges  on  the  two  plates 
disappear. 

148.  Relation  between  Electrostatic  Potential  and  Electric 
Potential.  —  The  drop  of  electrostatic  potential  between  any 
two  points  has  been  defined  as  the  work  done  by  the  agents  pro- 
ducing the  electrostatic  field  when  a  unit  positive  point-charge  is 


262  ELECTRICAL  ENGINEERING 

moved  from  one  point  to  the  other,  and  the  drop  of  electric  potential 
between  any  two  points  as  the  work  done  by  a  continuous  current 
in  an  insulated  conductor  connecting  the  two  points  per  unit 
quantity  of  electricity  transferred  across  each  section  of  the  conductor. 
Hence,  when  as  the  unit  of  charge  is  taken  the  electromagnetic 
unit  as  above  defined  and  the  erg  is  taken  as  the  unit  of  work, 
the  unit  of  electrostatic  potential  drop  is  the  number  of  ergs  of 
work  done  when  one  electromagnetic  unit  of  charge  is  moved 
from  one  point  to  the  other;  this  unit  is  called  the  electromag- 
netic unit  of  electrostatic  potential  difference.  Similarly,  when 
the  electrostatic  unit  is  taken  as  the  unit  of  quantity  of  electricity 
and  the  erg  as  the  unit  of  work,  the  unit  of  electric  potential 
drop  is  the  number  of  ergs  of  work  done  when  one  electrostatic 
unit  of  quantity  is  transferred  across  each  section  of  the  con- 
ductor ;  this  unit  is  called  the  electrostatic  unit  of  electric  potential 
difference.  The  relations  between  these  various  units  of  potential 
difference  are  given  in  Article  87. 

In  general,  every  source  of  electromotive  force,  either  contact 
or  induced,  produces  an  electrostatic  field  and  electric  charges 
appear  on  any  conductors  in  the  vicinity.  The  results  of  all 
known  experiments  show  that  the  drop  of  electrostatic  potential 
through  the  dielectric  between  any  two  points  which  are  connected 
by  conductors  is  exactly  equal  to  the  drop  of  electric  potential  through 
the  conductors  between  these  points,  provided  both  potential  drops 
are  expressed  in  the  same  units.  For  example,  in  the  experiment 
considered  above,  when  the  momentary  currents  in  the  galvanom- 
eters cease,  the  plate  connected  to  the  positive  terminal  of  the 
generator  must  be  at  the  same  electric  potential  as  this  terminal 
(this  follows  from  Ohm's  Law) ;  similarly,  the  plate  connected  to 
the  negative  terminal  must  be  at  the  same  electric  potential  as  this 
latter  terminal.  Hence  the  difference  of  electric  potential  between 
the  two  plates  is  equal  to  the  electromotive  force  of  the  generator, 
which  electromotive  force  is  readily  measured  by  any  of  the 
methods  described  in  Chapter  V.  The  difference  of  electrostatic 
potential  between  the  two  plates  can  be  determined  by  the  method 
given  in  Article  146.  When  the  values  of  the  potential  drops 
obtained  from  these  two  measurements  are  expressed  in  the  same 
units  they  are  found  to  be  equal. 

When  the  drop  of  potential  through  the  conductor  is  due  solely 
to  its  resistance,  i.e..  when  there  is  no  induced  or  contact  electro- 


ELECTROSTATICS  263 

motive  force  in  the  conductor,  we  have  seen  (Article  102)  that  the 
product  of  the  specific  resistance  of  the  conductor  by  the  current 
density  at  any  point  gives  the  drop  of  electric  potential  per  unit 
length  at  this  point  along  the  stream  lines  of  the  electric  current, 
which  drop  of  potential  per  unit  length  was  defined  as  the  electric 
intensity  at  the  point  in  question.  Since  the  electrostatic  in- 
tensity at  any  point  is  equal  to  the  drop  of  electrostatic  potential 
per  unit  length  measured  along  the  line  of  force  through  this  point 
(this  follows  directly  from  the  definition  of  electrostatic  potential, 
Article  144) ,  it  follows  that  the  electric  intensity  and  the  electrostatic 
intensity  are  the  same,  provided  they  are  both  expressed  in  the 
same  units.  Since  the  specific  resistance  of  ordinary  insulators 
is  very  high  the  current  density  in  them  is  very  small  for  such 
potential  differences  as  are  used  in  practice. 

149.  Displacement  Current.  —  When  the  two  plates  of  the 
electrometer  are  charged  by  connecting  them  to  the  two  terminals 
of  a  generator  or  other  source  of  electromotive  force,  we  have 
seen  that  a  momentary  electric  current  is  established  through  the 
wires  connecting  the  plates  to  the  terminals,  and  that  this  current 
is  in  the  direction  away  from  the  plate  which  becomes  negatively 
charged  and  towards  the  plate  which  becomes  positively  charged. 
In  the  case  of  an  electric  current  which  does  not  vary  with  time 
we  have  seen  that  the  path  of  the  current  is  a  continuous  line 
which  forms  a  closed  loop  lying  wholly  within  the  conductors 
forming  the  circuit.  The  question  arises  whether  the  path  of 
the  variable  current  which  results  in  the  establishment  of  an 
electric  charge,  such  as  that  produced  on  the  plates  of  the  elec- 
trometer, is  not  also  closed  on  itself ;  that  is,  whether  a  momentary 
current  is  not  also  established  through  the  dielectric  between  the 
plates.  To  determine  this,  it  is  necessary  to  see  whether  a  mag- 
netic field  due  to  some  external  source  produces  a  force  on  the 
dielectric  while  the  variable  current  exists  in  the  conducting 
part  of  the  circuit.  (See  definition  of  the  measure  of  an  electric 
current,  Article  66.)  To  determine  this  by  direct  experiment 
is  extremely  difficult,  though  it  has  been  done.  In  addition, 
the  results  of  all  known  experiments  are  in  accord  with  the  assump- 
tion that  an  electric  current  always  exists  in  the  dielectric  sur- 
rounding a  conductor  when  a  variable  current  exists  in  the  con- 
ductor; that  is,  when  the  charge  on  the  conductor  is  varying  there  is 
a  current  in  the  dielectric  continuous  with  the  current  in  the 


234  ELECTRICAL   ENGINEERING 

conductor.  There  are  two  important  differences  between  the 
properties  of  an  electric  current  which  can  exist  in  a  perfect 
dielectric  and  the  properties  of  a  current  in  a  conductor.  (1) 
The  current  in  a  conductor  may  be  "  continuous,"  that  is,  not 
varying  with  time,  or  it  may  be  variable ;  in  a  perfect  dielectric, 
however,  only  a  variable  current  can  exist.  (2)  The  current  in 
a  perfect  dielectric  does  not  develop  heat  energy,  while  in  a  con- 
ductor heat  energy  is  always  produced.  In  an  imperfect  insulator 
both  kinds  of  currents  may  exist;  that  part  of  the  total  current 
which  develops  heat  energy  due  to  the  resistance  of  the  dielectric 
is  called  the  conduction  current,  while  that  part  which  depends 
upon  the  time  variation  of  the  electrostatic  field,  and  therefore 
upon  the  time  variation  of  the  charges  on  the  conductors  in 
contact  with  the  dielectric,  is  called  the  displacement  current. 

In  the  case  of  a  conductor  in  contact  with  a  perfect  insulator, 
the  charge  given  any  element  of  the  surface  of  the  conductor  in 
any  small  interval  of  time  dt  is  equal  to  the  quantity  of  electricity 
dq=idt  transferred  to  that  surface  by  the  current  i  in  the  con- 
ductor provided  charge  and  current  are  measured  in  the  same 
system  of  units ;  hence,  since  in  this  case  the  only  current  leaving 
that  surface  is  the  displacement  current  in  the  dielectric,  it  follows 
that  the  displacement  current  leaving  that  surface  is 

.  =  dq  (17) 

dt 

That  is,  the  displacement  current  leaving  the  surface  of  a  conductor 
in  contact  with  a  perfect  dielectric  is  equal  to  the  time  rate  of  change 
of  the  charge  on  that  surface.  When  the  dielectric  in  contact  with 
the  conductor  is  not  a  perfect  insulator,  part  of  the  conduction 
current  coming  up  through  the  conductor  to  its  surface  leaves 
that  surface  as  a  conduction  current  through  the  dielectric. 
Hence  calling  id  the  displacement  current  in  the  dielectric  and  ic 
the  conduction  current  in  the  dielectric,  we  have  that  the  total 
current  coming  up  to  the  surface  at  any  instant  is 

i=id+ic  (18) 

that  is,  the  total  current  coming  up  to  a  surface  at  any  instant  is 
equal  to  the  total  current  leaving  that  surface  at  that  instant. 

Since  id  =  — ,  where  —  is  the  time  rate  of  change  of  the  charge 
dt  dt 

on  the  surface,  we  have  that 


ELECTROSTATICS  265 

dq=(i-ic)  dt 
whence  q=f'  (i-ic)  dt  (19) 

J  o 

where  q  is  the  total  charge  given  to  the  surface  in  any  interval 
of  time  t.  That  is,  the  total  charge  given  the  surface  of  a  conductor 
in  any  interval  of  time  is  equal  to  the  algebraic  sum  of  the  quantities 
of  electricity  transferred  to  that  surface  in  this  interval  by  the  con- 
duction currents  flowing  toward  this  surface. 

Since  the  electrostatic  flux  density  at  a  point  just  outside  a 
charged  conductor  is  B=4Trcr,  where  cr  is  the  surface  density 
of  the  charge  on  the  conductor,  it  follows  that  the  current  density 
of  the  displacement  current  just  outside  a  charge  conductor, 
which  must  be  equal  to  the  rate  of  change  of  the  surface  density 
of  the  charge  on  the  conductor,  is 

do-  _     1   dB  (20) 

dt        4^TT~dt 

The  results  of  all  known  experiments  show  that  —   —  is  a 

47T    dt 

perfectly  general  expression  for  the  current  density  of  the  displace- 
ment current  at  any  point  in  a  dielectric,  when  B  represents  the 

D 

electrostatic    flux   density    at   that    point.      The   quantity  —  is 

4?r 

.sometimes  called  the  electric  displacement;  it  represents  the  quan- 
tity of  electricity  transferred  by  the  displacement  current,  or  dis- 
placed, across  unit  area  perpendicular  to  the  direction  of  the 
electrostatic  flux  density. 

The  reader  may  find  it  difficult  at  first  to  grasp  the  idea  of 
displacement  current  in  an  insulator,  but  a  re-reading  of  Article 
64  will  make  evident  how  such  a  variable  current  might  occur 
in  a  dielectric.  If  electricity  has  the  property  of  an  incom- 
pressible fluid  filling  all  space,  then  what  is  called  a  positive  charge 
on  a  conductor  represents  a  displacement  of  electricity  in  one 
direction  across  the  surface  of  separation  between  a  conductor 
and  a  dielectric,  and  what  is  called  a  negative  charge  on  a  con- 
ductor represents  a  displacement  of  electricity  across  this  surface 
in  the  opposite  direction.  A  corresponding  displacement  of 
electricity  occurs  in  the  dielectric  and  also  in  the  conductor, 
but  in  the  case  of  a  dielectric  the  amount  of  displacement  is  limited 
by  a  property  of  the  dielectric  analogous  to  the  elastic  property 
of  a  solid,  such  as  that  forming  the  walls  of  the  cellular  structure 


266  ELECTRICAL  ENGINEERING 

described  in  Article  64,  while  a  conductor  has  no  such  elastic 
property.  What  is  called  a  charged  surface  is  then  analogous 
to  the  surface  of  separation  between  two  bodies  which  are  dif- 
ferently strained. 

150.  Dielectric  Strength.  —  Discharge  from  Points.  —  Corona. 
—  Since  the  quantity  of  electricity  displaced  across  unit  area 
in  a  dielectric  is  proportional  to  the  electrostatic  flux  density  at 
this  area,  which  in  turn  is  proportional  to  the  electrostatic  inten- 
sity at  this  point,  the  strain  produced  in  the  dielectric  at  any  point 
may  be  looked  upon  as  proportional  to  the  field  intensity  at  that 
point.  It  is  found  by  experiment  that  when  the  electrostatic 
intensity  at  any  point  in  a  dielectric  is  increased  up  to  a  certain 
definite  value,  the  dielectric  is  ruptured  at  this  point,  as  is  evi- 
denced by  the  spark  which  occurs,  which  in  general  extends 
through  the  dielectric  from  one  of  the  charged  conductors  to  the 
other.  This  is  analogous  to  rupture  of  any  substance  when  it  is 
mechanically  strained  up  to  its  breaking  point.  The  value  of  the 
electrostatic  intensity  which  causes  the  rupture  of  a  dielectric 
is  called  the  dielectric  strength  of  the  dielectric. 

Recent  experiments  show  that  the  dielectric  strength  of  or- 
dinary insulators,  such  as  rubber,  mica,  porcelain,  cloth,  etc., 
depends  not  only  upon  the  chemical  nature  of  the  insulator  but 
also  upon  its  thickness.  The  dielectric  strength  of  a  thin  piece 
of  insulation  is  greater  than  the  dielectric  strength  of  a  thick 
piece;  also,  several  thin  layers  making  a  given  thickness  have  a 
greater  dielectric  strength  than  a  single  layer  of  the  same  total 
thickness.  In  engineering  work  dielectric  strength  is  usually 
expressed  as  so  many  volts  per  inch;  see  Article  102. 

Since  the  electrostatic  intensity  just  outside  a  conductor  is 
proportional  to  the  surface  density  of  the  charge  on  the  conductor 
(see  equation  11)  and  since  the  surface  density  in  turn  is  in  general 
greatest  at  sharp  points  in  the  surface  of  a  conductor,  the  di- 
electric as  a  rule  breaks  down  first  at  the  sharpest  point  of  the 
conductor.  The  protective  effect  of  a  lightning  rod  is  based 
upon  this  fact. 

In  the  case  of  a  non-uniform  electrostatic  field,  such  as  the 
field  about  the  wires  of  a  transmission  line  or  in  the  insulation 
of  a  cable,  the  field  intensity  under  certain  conditions  can  exceed 
the  dielectric  strength  of  the  insulation  in  the  immediate 
vicinity  of  the  wire  without  exceeding  the  dielectric  strength  in 


ELECTROSTATICS  267 

the  rest  of  the  insulation.  Under  these  conditions  the  part  of 
the  insulation  where  the  dielectric  strength  is  exceeded  apparently 
becomes  a  fairly  good  conductor.  In  the  case  of  gaseous  and 
liquid  insulators  this  change  in  the  nature  of  the  insulation  is 
accompanied  by  the  appearance  of  a  bluish  light  in  the  vicinity 
of  the  wire.  The  region  around  the  wire  in  which  this  change 
takes  place  is  called  the  "  corona,"  and  the  whole  phenomenon 
is  called  the  "  corona  effect." 

151.  Electric  Capacity.  —  Electric  Condenser.  —  When  the  elec- 
trostatic potential  drop  along  the  lines  of  electrostatic  induction 
varies  with  time,  there  is  also  a  variation  of  the  electrostatic 
flux  density,  and,  as  noted  in  Article  149,  this  variation  of  the 
electrostatic  induction  with  respect  to  time  gives  rise  to  an 
electric  current  (i.e.,  the  displacement  current).  The  numerical 
value  of  the  ratio  of  the  displacement  current  through  a  given 
portion  of  a  dielectric  between  any  two  equipotential  surfaces 
to  the  time  rate  of  change  of  the  electrostatic  potential  difference 
between  these  two  surfaces  is  denned  as  the  electric  capacity  of 
this  portion  of  the  dielectric.  That  is,  when  the  displacement 
current  in  the  given  portion  of  the  dielectric  is  i  and  the  time  rate 

of  change  of  the  electrostatic  potential  difference  is  — ,  the  capacity 

di 

of  the  given  portion  of  the  dielectric  is 

C=± 

dv  (21) 

dt 

Compare  with  the  definition  of  self  inductance  of  a  circuit,  Article 
116.  When  the  current  is  expressed  in  abamperes,  the  potential 
difference  in  ab volts  and  the  time  in  seconds,  the  unit  of  capacity 
is  called  the  abfarad;  when  these  quantities  are  expressed  in 
amperes,  volts  and  seconds  respectively  the  unit  of  capacity  is  the 
farad.  When  the  current  is  expressed  in  c.  g.  s.  electrostatic  units 
(i.e.,  electrostatic  units  of  quantity  per  second),  the  potential 
difference  in  electrostatic  units  and  the  time  in  seconds,  the 
capacity  is  said  to  be  so  many  electrostatic  units.  Capacities 
ordinarily  dealt  with  are  of  the  order  of  one-millionth  of  a  farad  ; 
hence  the  micro-farad,  equal  to  one-millionth  of  a  farad,  is  also 
used  as  a  unit  of  capacity.  The  units  are  related  to  one  another 
as  follows: 


268  ELECTRICAL  ENGINEERING 

1  abfarad  =  109  farads 
1  abfarad  =  1015  microfarads 
1  abfarad=9X  1020  c.  g.  s.  electrostatic  units 
When  the  equipotential  surfaces  between  which  the  displace- 
ment current  flows  are  the  surfaces  of  conductors,  the  dielectric 
and  the  conductors  forming  these  surfaces  are  said  to  form  an 
electric  condenser.     The  capacity  of  a  condenser  formed  by  two 
conducting  surfaces  and  a  dielectric  may  be  expressed  in  terms 
of  the  charge  on  either  conductor  and  the  difference  of  potential 
between  them.     When  the  conducting  surfaces  are  such  that  the 
displacement  current  leaving  the  first  surface  is  the  same  as  that 
entering  the  second,  the  charges  on  the  two  surfaces  must  be  equal 
and  opposite,  see  equation  (19),  and  therefore  the  displacement 

current  is  — ,  where  q  is  the  numerical  value  of  the  charge  at  any 
dt 

instant  on  either  conductor,  see  equation  (17).  Hence  the  capac- 
ity of  the  condenser  formed  by  the  dielectric  and  the  two  con- 
ductors is 

C=JL  J-i 

dv    dv  (22) 

dt 

Therefore  the  capacity  of  the  condenser  is  equal  to  the  numerical 
value  of  the  charge  on  either  surface  per  unit  difference  of  potential 
between  the  two. 

The  capacity  of  such  a  condenser  is  constant,  i.e.,  —  is  constant, 

dv 

and  equal  to  the  ratio  of  the  charge  q  at  any  instant  to  the  potential 
difference  v  at  this  instant,  provided  the  dielectric  constants  of 
every  body  in  the  field  are  constant.  This  may  be  proved  as 
follows:  The  electrostatic  intensity  H  produced  at  any  point  P 
when  a  conductor  is  charged  with  ql  units  of  electricity  depends 
in  general  upon  the  size  and  shape  of  the  conductor,  the  nature  of 
the  surrounding  bodies,  and  the  size  and  shape  of  these  bodies 
and  their  position  with  respect  to  the  charged  conductor.  For 
these  conditions  fixed,  the  field  intensity  due  to  the  charge  q, 
and  whatever  other  charges  it  may  induce  is  proportional  to  this 
charge  qt,  and  therefore  the  drop  of  potential  due  to  this  charge  q, 
between  any  two  points  in  the  field  is  proportional  to  qlt  that  is 


ELECTROSTATICS  269 

where  A^  is  a  constant  depending  on  the  size  and  shape  of  the 
charged  conductor,  the  size,  shape  arid  position  of  any  other 
conductors  in  the  vicinity,  and  the  nature,  size,  shape  and  position 
of  the  surrounding  dielectric  bodies,  but  is  independent  of  the 
value  of  g,.  Similarly  the  drop  of  potential  between  these  same 
two  points  due  to  any  other  charge  q2  is  proportional  to  q2,  that 
is 

v2  =  A2q2 

Hence  the  total  drop  of  potential  between  1  and  2  due  to  both 
charges  is 

v=vl  +  v2=Alql  +  A2q2 

By  definition,  however,  in  the  case  of  a  condenser  q2  =  —  ql,  and 
therefore  the  difference  of  potential  between  the  two  conductors 
may  be  written 

v=vl-v2=Alq2-  A2q2=(A,-  A2)  q} 
Whence,  from  (22),  the  capacity  of  the  condenser  is  a  constant 

equal  to  -  . 
A,-A2 

152.  Simple  Forms  of  Condensers.  —  a.  Parallel  Plate  Con- 
denser. —  The  parallel  plate  electrometer  described  in  Article 
146  is  one  of  the  simplest  forms  of  electric  condensers.  Its 
capacity  is  calculated  directly  from  equations  (16)  which  give 
the  relation  between  the  charge  on  each  plate  and  the  difference 
of  potential  between  them.  That  is,  the  capacity  of  such  a 
parallel  plate  condenser  is 


O 

C=  —  =  --  c.  g.  s.  electrostatic  units         (23) 

V 


where  K  is  the  dielectric  constant  of 
the  dielectric  between  the  plates,  S 
the  area  of  each  plate  and  D  the  dis- 
tance between  the  plates. 

b.  Spherical  Condenser.  —  Another 
simple  form  of  condenser  consists  of 
two  concentric  spherical  shells,  the 
space  between  which  is  filled  with  a 
uniform  dielectric.  Let  the  shells  have 
the  radii  rl  and  r2  and  let  the  dielectric 
constant  of  the  dielectric  between  the 
two  shells  be  K.  A  charge  of  Q  c.  g.  s.  electrostatic  units  given  the 
inside  sphere  will  induce  a  charge  of  —  Q  units  on  the  inside  surface 


270  ELECTRICAL  ENGINEERING 

of  the  outer  shell  and  a  charge  qf  +  Q  units  on  the  outside  surface 
of  this  shell  and  whatever  other  conductors  may  be  connected 
to  it.  This  outside  charge  +  Q,  however,  will  have  no  effect  on 
the  field  intensity  inside  this  shell  (see  Article  137). 

From  symmetry,  the  lines  of  electrostatic  induction  are  radial 
lines  normal  to  the  surfaces  of  two  spheres;  the  total  number 
coming  out  from  the  charge  Q  is  4  TT  Q  (see  Article  141).  Hence 
the  electrostatic  flux  density  at  any  point  P  in  the  dielectric  is 

B  = =—  where  r  is  the  distance  of  P  from  the  center  of  the 

4?r       T*  BO 

spheres.     Hence  the  field  intensity  at   P  is    H= —  =  — 1_    and 

therefore  the  difference  of  potential  between  the  two  spheres  is 
V  = 


r2 

Hence  the  capacity  of  this  condenser  is 

0       r  r 
C=—=-     —  K  c.  g.  s.  electrostatic  units        (24) 

c.  Coaxial  Cylinders.  —  Applying  exactly  similar  reasoning  to 
the  case  of  two  coaxial  cylinders  which  are  sufficiently  long  in 
comparison  with  their  diameters  so  that  the  lines  of  electrostatic 
induction  going  out  the  ends  may  be  neglected,  we  have  from 
symmetry  that  the  electrostatic  flux  density  at  any  point  P  in 

the    dielectric    between    the    two    cylinders    is    B  = —  = — — 

2  irr        r 

where  Q  is  the  charge  per  unit  length  of  the  condenser,  and  r  the 
distance  of  P  from  the  center  of  the  cylinders.     Hence  the  field 

2  Q 

intensity  at  P  isH= and  therefore  the  difference  of  potential 

Kr 


between  the  two  cylinders  is 


where  r,  is  the  outside  radius  of  the  inside  cylinder  and  r2  the 
inside  radius  of  the  outside  cylinder.  Hence  the  capacity  per 
centimeter  of  the  condenser  formed  by  two  coaxial  cylinders  is 

O          K 
C=—= c.  g.  s.  electrostatic  units  (25) 

V    2lnL> 


ELECTROSTATICS  271 

This  formula  is  applicable  to  a  single  wire  enclosed  in  a  lead 
sheath.     For  practical  calculations  this  formula  may  be  written 

C  =  —  microfarads  per  1000  feet       (25a) 

log-2 
r, 

or 

n     0.03883  A'  . 

C  =  —  microfarads  per  mile  (256) 

log-2 
r, 

d.  Two  Parallel  Cylinders.  —  In  case  of  two  parallel  cylinders 
which  are  not  coaxial,  Fig.  79,  but  a  distance  D  apart,  i.e.,  two 
parallel  wires,  an  approximate  formula  for  the  capacity  per  unit 

+Q  -Q 

p 
dx 


Fig.  79. 

length,  in  case  the  cylinders  are  long  compared  to  their  distance 
apart,  can  be  derived  by  assuming  the  charge  on  each  cylinder  is 
uniformly  distributed  over  the  surface  of  the  cylinder.  In  this 
case  the  electrostatic  intensity  at  any  point  P  on  the  line  joining 

2  Q 
the  centers  of  the  two  cylinders  due  to  the  charge  on  1  is  Hl  = — 

Kx 

and  the  electrostatic  intensity  at  P  in  the  same  direction  due  to 

2  Q 

the  charge  —  Q  on  2  is   H2= ;   therefore  the   total   in- 

K  (D  —  x) 

2 
between  1  and  2  is 


tensity  at  P  is  ~jr   "  +  7^7^  -    Hence  the  difference  of  potential 


4Q     D-r 


Hence  the  capacity  per  centimeter  of  the  condenser  formed  by  the 
two  parallel  wires  when  D  is  large  in  comparison  with  r  is  (ap- 
proximately) 


272 


ELECTRICAL  ENGINEERING 


K 


V       A    7     D 

4  In  — 


c.  g.  s.  electrostatic  units        (26) 


For  practical  calculations  this  formula  may  be  written 
0.003677  K 


or 


log  — 
r 


0.01941 

log  — 
r 


microfarads  per  1000  feet    (26a) 


microfarads  per  mile  (266) 


[Neither  —or  is  correct  when  the  wires  are  close  together, 

r          r 

since  the  charge  on  the  two  wires  is  no  longer  uniformly  dis- 
tributed (see  Alex.  Russell,  Alternating  Currents,  Vol.  I,  p.  99). 
When  the  wires  are  far  apart  r  is  .negligible  in  comparison  with 
D,  hence  formula  (26)  is  sufficiently  accurate.] 

153.  Specific  Inductive  Capacity.  —  In  all  the  formulas  for 
capacity  it  is  seen  that  the  capacity  varies  directly  as  the  dielec- 
tric constant  of  the  medium  between  the  two  conductors.  Hence 
by  measuring  the  capacity  Ca  of  a  given  condenser  when  the 
plates  are  separated  by  air  and  then  measuring  the  capacity  Ck 
when  the  plates  are  separated  by  any  other  dielectric,  the  dielec- 
tric constant  may  be  readily  determined  experimentally,  since 

C 
K= — -.     Since  the  numerical  value  of  the  dielectric   constant 

Ca 

is  equal  to  the  ratio  of  two  capacities,  this  constant  is  frequently 
C  C2  C8  called  the  "  specific  inductive 

capacity  "  of  the  dielectric. 

154.  Condensers  in  Series 
and  in  Parallel.  —  When  two 
or  more  condensers  are  con- 
nected in  series,  equal  and 
opposite  charges  will  be  in- 
Fig.  so.  '  duced  on  each  pair  of  plates 

connected  by  a  conductor.  Hence  if  the  condensers  have  capac- 
ities d,  d,  d,  etc.,  the  total  potential  drop  through  all  the  con- 
densers is 


j 

1 

i 

+Q 

-Q      -K 

-T 

Q       i-C 

r  

r 

Q          j 

ELECTROSTATICS 


Q     Q     Q      Vi      i      A 

=c,+ca+c3=Q  (c+c+cj 


273 


Hence  the  equivalent  capacity  of  any  number  of  condensers  in 
series  is  Cs  where 


Cs     Q     C,     C2     C3 

When  the  condensers  are  connected  in  parallel,  the  drop  of 
potential  across  each  condenser  is  the  same,  and  therefore  the 


Fig.  81. 

total  positive  charge  given  all  the  condensers  (equal  to  the  total 
negative  charge)  is 

Hence  the  equivalent  capacity  of  any  number  of  condensers  in 
parallel  is 


c     Q 
= 


(28) 


Compare  with  resistances  in  series  and  parallel,  Article  98. 

155.  Absorption  in  a  Condenser.  —  Residual  Charge.  —  Ex- 
periment shows  that  when  a  given  difference  of  potential  is  estab- 
lished between  the  plates  of  a  condenser  the  dielectric  of  which 
is  a  solid,  the  charge  taken  by  the  condenser  depends  upon  the 
length  of  time  this  potential  difference  is  maintained.  Again, 
when  a  charged  condenser  is  discharged  by  connecting  its  plates 
momentarily  with  a  conductor,  and  this  connection  is  then  broken, 
the  plates  at  first  appear  to  be  entirely  discharged,  but  after  a 
few  seconds  a  charge  again  appears  on  them  resulting  in  the  re- 
establishment  of  a  difference  of  potential  (less  than  in  the  first 
case)  between  the  plates.  In  short,  a  solid  dielectric  apparently 
absorbs  a  certain  amount  of  charge  which  it  gives  up  only  after 
a  considerable  lapse  of  time.  The  charge  which  appears  on  the 
plates  of  the  condenser  after  the  first  discharge  is  called  the 
"  residual  "  charge ;  this  phenomenon  is  called  electric  absorption. 
The  absorption  of  a  dielectric  is  apparently  due  to  impurities  in 


274  ELECTRICAL  ENGINEERING 

it  or  to  lack  of  homogeneity.  It  is  greatest  in  substance  like  mica 
and  glass ;  it  is  doubtful  if  absorption  occurs  at  all  in  absolutely 
homogeneous  substances. 

156.  Energy  of  an  Electrostatic  Field.  —  An  electrostatic  field 
represents  a  certain  amount  of  stored  energy  just  as  a  magnetic 
field  represents  a  certain  amount  of  stored  energy.  Let  a  con- 
denser be  charged  by  being  connected  to  a  battery  or  other  source 
of  electromotive  force;  the  amount  of  energy  stored  in  the  elec- 
trostatic field  can  be  readily  calculated.  Let  the  capacity  of  the 
condenser  be  C  and  let  the  drop  of  electrostatic  potential  from 
its  positive  to  its  negative  plate  at  any  instant  be  v.  Then  the 
charge  on  each  plate  of  the  condenser  at  this  instant  is  numerically 
equal  to  Cv.  To  change  the  charge  by  an  amount  dq  =  Cdv  in 

time  dt  requires  a  current  of  —  =  C —  units  for  dt  seconds.     In 

dt         dt 

the  battery  and  the  wires  connecting  the  battery  to  the  plates  of 
the  condenser  this  current  will  be  in  "the  direction  from  the  nega- 
tive to  the  positive  plate  of  the  condenser  and  in  the  dielectric 
separating  the  plates  of  the  condenser  this  current  (as  a  displace- 
ment current)  will  be  in  the  direction  from  positive  to  the  nega- 
tive plate  of  the  condenser,  that  is  in  the  direction  of  the  drop  of 
potential  through  the  condenser.  Hence  an  amount  of  energy 

dv 

equal  to  C— .  v.  dt=Cvdv  will  be  lost  by  the  current  in  the  con- 
dt 

denser.  Hence,  when  the  drop  of  potential  between  the  two 
plates  of  the  condenser  changes  by  an  amount  dv  an  amount  of 
energy  equal  to  Cvdv  is  gained  by  the  electrostatic  field  in  the 
dielectric  separating  the  two  plates.  Therefore,  when  the  differ- 
ence of  potential  between  the  two  plates  is  increased  from  zero  to 
v  a  total  amount  of  energy 


J, 


:-CV  = — ±-=-qv  (29) 

O  C\      /°f  O      i  \    .     / 

0 

is  stored  in  the  electrostatic  field.  In  using  these  formulas,  care 
must  be  taken  to  express  all  the  quantities  in  the  same  system 
of  units. 

The  energy  which  is  stored  in  the  electrostatic  field  comes 
from  the  source  of  electromotive  force  to  which  the  condenser  is 
connected.  When  the  plates  are  disconnected  from  the  source 
of  electromotive  force  and  are  connected  to  each  other  by  a  wire. 


ELECTROSTATICS  275 

the  condenser  discharges  and  the  energy  which  was  stored  in  the 
electrostatic  field  is  transferred  to  the  wire  as  heat  energy.  A 
charged  condenser  is  then  the  seat  of  an  electromotive  force,  and 
the  direction  of  this  electromotive  force  through  the  condenser  is 
always  in  the  direction  from  the  negative  plate  of  the  condenser 
to  the  positive  plate.  Hence,  when  the  condenser  is  being  charged 
its  electromotive  force  opposes  the  current,  that  is,  the  electro- 
motive force  of  the  condenser  is  a  back  electromotive  force ;  when 
the  condenser  discharges,  the  electromotive  force  is  in  the  direc- 
tion of  the  current. 

It  can  be  shown  by  a  similar  process  of  reasoning  to  that 
employed  in  Article  125  that  the  total  energy  of  any  system  of 
charged  conductors  may  be  expressed  as  the  volume  integral 


=J-f    R&d^J-f    * 

8?rj  y  StrJ  v  K 


W=—         KH2  dv=—         —  dv          (30a) 

STTJ  y  STrJ  v  K 

throughout  all  space,  where  dv  is  any  elementary  volume,  H  the 
electrostatic  intensity  at  dv,  B  the  electrostatic  induction  at  dv 
and  K  the  dielectric  constant  of  the  dielectric  in  the  space  occu- 
pied by  dv.  Or,  more  briefly,  the  energy  per  unit  volume  of  the 
electrostatic  field  is 

KH2        B2  (306) 


8?r        8-n-K 

157.  Dielectric  Hysteresis.  —  No  substance  is  a  perfect  dielec- 
tric, but  even  the  best  insulators,  such  as  porcelain  or  glass,  are 
conductors  to  a  slight  extent.  Hence  when  a  constant  electro- 
motive force  is  impressed  across  a  condenser,  a  small  continuous 
current  is  established  through  the  dielectric  and  therefore  a  cer- 
tain amount  of  heat  energy  is  dissipated  in  the  dielectric.  When 
the  potential  drop  across  the  condenser  is  V  and  the  conductance 
of  the  condenser  is  g,  the  rate  at  which  heat  energy  is  developed 
is  (Joule's  Law)  equal  to  gV2.  When  an  alternating  potential 
difference  (see  Chapter  VI)  of  the  same  effective  value  is  estab- 
lished across  the  condenser  it  is  found  that,  when  the  dielectric 
of  the  condenser  possesses  the  property  of  electric  absorption,  the 
heat  energy  dissipated  in  the  condenser  is  greater  than  gV2.  This 
increase  in  the  heat  energy  dissipated  in  the  dielectric  is  some- 
times said  to  be  due  to  dielectric  hysteresis.  The  phenomenon, 
however,  is  probably  not  due  to  a  lag  of  the  flux  density  behind 
the  resultant  field  intensity,  as  in  the  case  of  magnetic  hysteresis, 


276  ELECTRICAL    ENGINEERING 

but  is  rather  due  to  the  existence  in  the  dielectric  of  small  con- 
ducting particles,  insulated  from  one  another,  which  form  minute 
condensers  and  in  which  the  varying  electrostatic  field  sets  up 
alternating  currents  of  greater  magnitude  than  would  be  set  up 
by  a  constant  impressed  electromotive  force.  The  increase  of 
heat  energy  dissipated  when  an  alternating  potential  difference  is 
established  across  the  condenser  over  that  dissipated  when  a  con- 
stant potential  difference  is  established,  is  then  due  to  the  excess 
of  heat  energy  developed  by  the  alternating  currents  in  these 
conducting  particles. 

SUMMARY  OF  IMPORTANT  DEFINITIONS  AND 

PRINCIPLES 

(Note:  All  formulas   are   in   c.  g.  s.   electrostatic  units  unless 
otherwise  specified.) 

1.  An  electrostatic  unit  point-charge  is  a  charge  which  repels 
with  a  force  of  one  dyne  an  equal  point-charge  one  centimeter 
away. 

2.  Two  point-charges  of  q  and  cf  units  at  a  distance  r  centi- 
meters apart  repel  each  other  with  a  force  of 


3.  The  electrostatic  field  intensity  H  at  any  point  in  an  electro- 
static field  is  the  force  in  dynes  which  would  act  on  a  unit  positive 
point-charge  place  at  that  point  due  solely  to  the  agents  producing 
the  original  field. 

4.  The  field  intensity  at  any  point  due  to  a  point-charge  q  at 
a  distance  r  centimeters  away  is 


5.  The  mechanical  force  exerted  on  a  point-charge  q  is 

F=qH 

where   H  is  the  field  intensity  due  to  every  agent  in  the  field 
except  the  charge  q. 

6.  Lines  of  electrostatic  force  are  lines  drawn  in  an  electro- 
static field  in  such  a  manner  that  they  coincide  in  direction  at 
each  point  P  with  the  field  intensity  at  P  and  their  number  per 
unit  area  at  each  point  P  across  a  surface  at  right  angles  to  their 
direction  is  equal  to  the  field  intensity  at  P. 


ELECTROSTATICS  277 

7.  Gauss's  Theorem.  —  The  algebraic  sum  of  the  number  of 
lines  of  electrostatic  force  outward  from  any  closed  surface  is  equal 
to  4  TT  times  the  algebraic  sum  of  the  charges  inside  this  surface. 

8.  A  dielectric  in  an  electrostatic  field  is  considered  to  be  made 
up  of  filaments  such  that  were  the  lateral  walls  of  any  one  of  these 
filaments  separated  from  the  rest  of  the  dielectric  by  a  narrow 
air  gap,  no  charges  would  be  induced  on  these  walls. 

9.  The  intensity  of  electrisation  /  at  any  point  in  a  body  is  de- 
fined as  the  charge  per  unit  area  which  would  appear  on  a  gap 
cut  in  the  body  at  this  point  perpendicular  to  the  line  of  electrisa- 
tion through  this  point.     The  direction  of  the  intensity  of  electri- 
sation is  the  direction  of  the  line  of  electrisation,  the  positive  sense 
of  which  is  the  sense  of  the  line  drawn  into  the  gap  from  the  wall 
on  which  the  positive  charge  is  formed. 

10.  A  line  of  electrisation,  or  unit  filament,  is  a  filament  of  such 
a  size  that  were  it  broken  by  a  narrow  gap  at  any  point,  the 
numerical  value  of  the  charge  formed  on  either  wall  of  the  gap 

would  be  — -. 

47T 

11.  The  number  of  lines  of  electrostatic  induction,  or  flux  of 
electrostatic  induction,  crossing  any  surface  is  defined  as  the  alge- 
braic sum  of  the  number  of  lines  of  electrisation  and  the  lines  of 
electrostatic  force  crossing  that  surface. 

12.  The  electrostatic  flux  density  B  at  any  point  is  the  number 
of  lines  of  electrostatic  induction  per  square  centimeter  crossing 
an  elementary  surface  drawn  at  this  point  normal  to  their  direc- 
tion.    The  electrostatic  flux  density  is  the  vector  sum 

B  =  47rJ+H 

13.  In  an  electrostatic  field  due  solely  to  electric  charges,  a 
line  of  electrostatic  force  originates  at  a  positive  charge  and  ends 
at  a  negative  charge,  and  may  exist  in  any  dielectric  but  not  in  a 
conductor  in  which  no  current  is  flowing.     A  line  of  electrisation 
originates  at  a  negative  induced  charge  on  a  dielectric  and  runs 
through  the  dielectric  to  a  positive  induced  charge  on  the  surface 
of  the  dielectric.     A  line  of  electrostatic  induction  is  a  continuous 
line  which  originates  at  a  positively  charged  conductor  (or  at  a 
positive  contact  charge  on  a  dielectric)  and  ends  at  a  negatively 
charged  conductor  (or  at  a  negative  contact  charge  on  a  dielectric), 
but  passes  through  the  induced  charges  on  dielectrics. 

14.  The  dielectric  constant,  or  specific  inductive  capacity,  K,  of 


278  ELECTRICAL  ENGINEERING 

a  dielectric  is  the  ratio  of  the  electrostatic  flux  density  B  in  the 
dielectric  to  the  resultant  electrostatic  intensity  H  when  the 
dielectric  is  placed  in  an  electrostatic  field  ;  that  is 

K=* 
H 

15.  A  closed  hollow  conductor  completely  screens  everything 
inside  from  external  electrostatic  influences. 

16.  The  electrostatic  intensity  within  the  substance  of  a  con- 
ductor in  which  there  is  no  electric  current  is  zero. 

17.  The  total  charge  within  any  region  completely  enclosed  by 
a  conductor  is  zero. 

18.  The  electrostatic  intensity  at  any  point  P  just  outside  a 
charged  conductor  in  which  there  is  no  electric  current  is  normal 
to  the  surface  and  equal  to 


K 

where  K  is  the  dielectric  constant  of  the  dielectric  in  contact  with 
the  conductor  at  P  and  cr  is  the  surface  density  of  the  charge  on 
the  conductor  just  opposite  P. 

19.  The  electrostatic  flux  density  at  any  point  P  just  outside 
a  charged  conductor  in  which  there  is  no  electric  current  is  normal 
to  the  surface  and  equal  to 

B  =4770- 

independent  of  the  dielectric  in  contact  with  the  conductor;  cr  is 
the  surface  density  of  the  charge  on  the  conductor  just  opposite  P. 

20.  The  electrostatic  potential  V  at  any  point  in  an  electrostatic 
field  is  the  work  which  would  be  done  by  the  agents  producing 
the  field  in  moving  a  unit  positive  point-charge  from  this  point  to 
infinity.     The  electrostatic  potential  at  any  point  due  to  a  point- 
charge  q  at  a  distance  r  centimeters  away  is 


The  resultant  electrostatic  potential  due  to  any  number  of  charges 
is  the  algebraic  sum  of  the  potential  due  to  all  the  individual  poles. 
21.  The  drop  of  electrostatic  potential  along  any  path  from 
any  point  1  to  any  point  2  is 

¥,-¥,=  A'  (Hcos0)dl 


where  dl  is  the  length  of  any  element  of  the  path  from  1  to  2,  H 
the  field  intensity  at  dl  and  0  the  angle  between  the  direction  of 


ELECTROSTATICS  279 

H  and  dl.  When  the  field  is  due  solely  to  electrostatic  charges 
the  drop  of  potential  is  independent  of  the  path  from  1  to  2. 
The  drop  of  electrostatic  potential  around  a  closed  path  is  equal 
to  the  algebraic  sum  of  the  contact  and  induced  electromotive 
forces  in  this  path,  provided  all  quantities  are  expressed  in  the 
same  system  of  units. 

22.  An  electrostatic  equipotential  surface  is  a  surface  drawn  in 
an  electrostatic  field  in  such  a  manner  that  the  drop  of  electro- 
static potential  along  any  path  in  the  surface  is  zero.     Such  a 
surface  is  perpendicular  at  each  point  to  the  line  of  electrostatic 
force  through  that  point. 

23.  When  as  the  unit  of  electrostatic  charge  is  taken  a  unit 
equal  to  3  X  1010  times  the  size  of  an  electrostatic  unit,  called  the 
electromagnetic  unit  of  charge,  the  charge  produced  at  a  conduct- 
ing surface  by  a  variable  current  is  equal  to  the  quantity  of  elec- 
tricity  (Article  80)  transferred  to  that   surface  by  the  electric 
current  in  the  conductor. 

24.  When  as  the  unit  of  electrostatic  potential  difference  is 

taken  a  unit  equal  to  times  the  size  of  the  electrostatic 

3X1010 

unit  of  potential  difference,  called  the  electromagnetic  unit  of 
electrostatic  potential  difference,  the  electrostatic  difference  of 
potential  between  any  two  points  is  equal  to  the  difference  of 
electric  potential  (Article  87)  in  ab volts  between  these  points. 

25.  When  as  the  unit  of  electrostatic  intensity  is  taken  a  unit 

equal  to times  the  size  of  the  electrostatic  unit  of  intensity, 

3X1010 

called  the  electromagnetic  unit  of  electrostatic  intensity,  the  electro- 
static intensity  at  any  point  is  equal  to  the  electric  intensity 
(Article  102)  in  abvolts  per  centimeter  at  this  point. 

26.  The  displacement  current  id  flowing   away  from  a  con- 
ducting surface  through  the  dielectric  in  contact  with  this  surface 
is  equal  to  the  time  rate  of  change  of  the  charge  q  on  this  surface, 
i.e., 

i,-dq 
d~dt 

where  all  quantities  are  in  the  sam*  system  of  units.  The  current 
density  of  the  displacement  current  at  any  point  in  a  dielectric  is 

_L  *!L 

4^  ~dt 


280  ELECTRICAL  ENGINEERING 

where  B  is  the  electrostatic  flux  density  and  all  quantities  are  in 
the  same  system  of  units. 

27.  The  value  of  the  ratio  of  the  displacement  current  between 
two  equipotential  surfaces  in  a  dielectric  to  the  time  rate  of 
change  of  the  difference  of  potential  between  these  two  surfaces 
is  called  the  capacity  C  of  the  portion  of  the  dielectric  between  the 
two  surfaces.  The  displacement  current  through  a  given  portion 
of  a  dielectric  between  two  equipotential  surfaces  is  then 


where  v  is  the  difference  of  potential  between  the  two  surfaces. 
The  c.  g.  s.  electrostatic  unit  of  capacity,  the  electromagnetic  unit 
or  abfarad  and  the  practical  unit  or  farad  are  related  as  follows: 

1  abfarad  =  109  farads  =9X  1020  c.  g.  s.  electrostatic  units. 
When  the  two  equipotential  surfaces  are  conducting  surfaces,  the 
dielectric  and  the  two  conducting  surfaces  are  said  to  form  an 
electric  condenser.  The  capacity  of  a  condenser  is  also  equal  to 
the  numerical  value  of  the  ratio  of  the  charge  on  either  surface 
to  the  difference  of  potential  between  the  two. 

28.  The  capacity  of  the  condenser  formed  by  two  long  parallel 
wires  of  circular  cross  section  is 

„    0.01941  K  .      ,  .. 

C  =  —  microfarads  per  mile 

l°9~ 

where  K  is  the  dielectric  constant  of  the  surrounding  medium,  D 
the  distance  between  centers  of  wires  and  r  the  radius  of  each  wire. 

29.  The  resultant  capacity  of  two  or  more  condensers  in  series 
is  the  reciprocal  of  the  sum  of  the  reciprocals  of  the  various 
capacities,  i.e., 

1_1       1 

C=C~1  +  C2  + 

The  resultant  capacity  of  two  or  more  condensers  in  parallel 
is  the  sum  of  the  various  capacities,  i.e., 
(7=C1+C2+- 

30.  The  energy  stored  in  the  dielectric  of  a  condenser  is 

TJ7       !  rLJ       l(f       ! 

W  =-  Of  =-  -  =-  qv 

2  2  C     2 

where  C  is  the  capacity  of  the  condenser,  q  the  numerical  value  of 
the  charge  on  either  conductor,  and  v  the  difference  of  potential 


ELECTROSTATICS  281 

between  the  conductors.     These  formulas  hold  for  any  system  of 
units  provided  all  quantities  are  expressed  in  the  same  system. 

PROBLEMS 

1 .  A  circular  disc  20  inches  in  diameter  and  of  negligible  thick- 
ness is  charged  uniformly  on  both  sides  with  a  total  charge  of 
15X105  electrostatic  units.     (1)  What  is  the  intensity  of  the  elec- 
trostatic field  at  a  point  on  a  normal  to  the  disc  at  its  center  10 
inches  from  the  plane  of  the  disc?    (2)  What  would  be  the  intensity 
at  this  point  if  the  disc  had  an  infinite  radius  and  the  surface 
density  of  the  charge  remains  the  same?    (3)  Under  the  latter  con- 
ditions what  would  be  the  difference  in  electrostatic  potential  be- 
tween the  point  and  the  disc? 

Ans.:  (1)1358  c.  g.  s.  electrostatic  units ;  (2)  4650  c.  g.  s.  electro- 
static units;  (3)  118,000  c.  g,  s.  electrostatic  units. 

2.  Two  plates  A  and  B,  the  areas  of  which  may  be  considered 
to  be  infinite,  are  placed  parallel  to  each  other  and  10  cm.  apart. 
A  is  charged  positively  with  50  electrostatic  units  of  charge  per 
sq.  cm.  while  B  is  charged  negatively  with  the  same  density;  the 
charges  are  confined  to  the  surfaces  which  face  each  other.     (1) 
Determine  the  pull  per  sq.  cm.  which  A  exerts  upon  B. 

A  slab  of  glass  of  infinite  area  and  8  cm.  in  thickness  is  now 
placed  between  the  two  plates  and  parallel  to  them,  but  not  touch- 
ing either.  The  dielectric  constant  of  the  glass  is  5.  (2)  Deter- 
mine again  the  pull  per  sq.  cm.  which  A  exerts  upon  B;  (3)  the 
intensity  of  electrisation  in  the  glass;  and  (4)  the  electrostatic  flux 
density  in  the  glass. 

Ans.:  (1)  15,700  dynes;  (2)  15,700  dynes;  (3)  40  c.  g.  s.  elec- 
trostatic units  ;  (4)  629  c.  g.  s.  electrostatic  units. 

3.  Determine  the  difference  of  electrostatic  potential  between 
the  two  plates  in  Problem  2  when  the  plates  are  charged  positively 
and  negatively  respectively,  with  0.05  c.  g.  s.  electrostatic  units 
per  sq.  cm.  (1)  when  the  medium  between  the  plates  is  air;  (2)  when 
a  slab  of  glass  of  infinite  area,  6  cm.  in  thickness  and  with  a  dielec- 
tric constant  equal  to  5,  is  placed  between  the  plates  and  parallel 
to  them.     The  charge  on  the  plates  and  the  distance  between  the 
plates  remain  constant  in  the  two  cases. 

If  the  difference  of  potential  between  the  two  plates  is  kept 
constant  at  0.1  electrostatic  unit,  -what  will  be  the  numerical  value 
of  the  charge  per  sq.  cm.  on  each  plate  (3)  when  the  medium  be- 


282  ELECTRICAL  ENGINEERING 

tween  the  plates  is  air;  (4)  when  the  slab  of  glass  6  cm.  thick  is 
placed  between  the  plates? 

Ans.:  (1)  6.29  c.  g.  s.  electrostatic  units ;  (2)  3.27  c.  g.  s.  electro- 
static units;  (3)  0.000795  c.  g.  s.  electrostatic  units;  (4)  0.00153 
c.  g.  s.  electrostatic  units. 

4.  Two  parallel  metallic  plates  each  100  sq.  cm.  in  area  are 
separated  by  a  distance  of  0.5  cm.     If  the  difference  of  potential 
between  the  two  plates  is  maintained  at  1000  volts,  determine 
(1)  the  charge  in  microcoulombs  on  each  plate  when  the  plates 
are  separated  by  air;  (2)  when  the  dielectric  between  the  two 
plates  is  glass;  (3)  the  work  required  to  pull  the  glass  out  of  the 
electrostatic  field.     The  dielectric  constant  of  glass  is  5.     Assume 
a  uniform  distribution  of  charge. 

Ans.:  (1)  0.01768  microcoulombs;  (2)  0.0884  microcoulombs; 
(3)  354  ergs. 

5.  Two  metallic  plates  each  100  sq.  cm.  in  area  are  charged 
until  the  difference  of  potential  between  the  plates  is  300  volts 
and  the  source  of  potential  is  then  removed.     The  plates  are  0.5 
cm.  apart  and  the  medium  between  them  is  air.     (1)  Determine 
the  work  done  in  separating  the  plates  until  they  are  one  centi- 
meter apart.     (2)  What  is  the  difference  of  potential  between  the 
plates  if  a  sheet  of  glass  0.5  cm.  thick  and  having  a  dielectric 
constant  of  5  is  then  pushed  in  between  them? 

Ans.:  (1)  7.96  ergs;  (2)  360  volts. 

6.  A  transmission  line  10  miles  in  length  consists  of  two  No. 
0000  wires  (B.  &  S.  gauge)  spaced  3  feet  between  centers.     If  a 
potential  difference  of  1000  volts  is  established  between  the  wires 
and  the  line  is  open  at  the  far  end,  determine  the  energy  in  the 
electrostatic  field  surrounding  the  line.     The  diameter  of  a  No. 
0000  wire  is  0.460  inch. 

Ans.:  0.0443  joules. 

7.  The  total  inductance  of  a  two-wire  transmission   line  25 
miles  in  length  is  84.6  millihenries.     What  is  its  capacity  in  micro- 
farads? 

Ans.:  0.223  microfarads. 

8.  Three  condensers,  A,  B  and  C,  the  capacities  of  which  are 
25,  20  and  15  microfarads  respectively,  are  connected  in  series. 
If  the  potential  drop  across  B  is  100  volts,  determine  (1)  the  po- 
tential drop  across  A;  (2)  the  potential  drop  across  C;  and  (3)  the 
potential  drop  across  the  three  condensers  in  series.     (4)  What  is 


ELECTROSTATICS  283 

the  charge  in  microcoulombs    on  each  condenser;    and  (5)  the 
capacity  of  the  three  condensers  in  series? 

Ans.:  (1)  80  volts;  (2)  133.3  volts;  (3)  313.3  volts;  (4)  2000 
microcoulombs ;  (5)  6.38  microfarads. 

9.  When  an  e.  m.  f.  of  200  volts  is  impressed  across  two  con- 
densers A  and  B  of  unknown  capacity  connected  in  series,  the 
potential  drop  through  A  is  three  times  that  through  B  and  the 
electrostatic  energy  of  the  system  is  0.5  joule.     What  would  be 
the  charge  upon  A  and  upon  B  when  an  e.  m.  /.  of  100  volts  is 
impressed  across  the  condensers  in  parallel? 

Ans.:  0.00333  and  0.01  coulomb  respectively. 

10.  A  lead-covered  cable  is  made  of  a  No.  00  B.  &  S.  wire  sur- 
rounded by  a  layer  of  rubber  0.25  inch  thick,  which  is  in  turn 
surrounded  by  a  layer  of  gutta-percha  0.25  inch  thick,  the  whole 
being  encased  in  the  lead  sheath.     The  specific  inductive  capacity 
of  the  rubber  and  the  gutta-percha  is  2.2  and  4.5  respectively. 
What  is  the  ratio  of  the  potential  drop  through  the  rubber  to  that 
through  the  gutta-percha  when  a  given  difference  of  potential  is 
established  between  the  wire  and  the  sheath?     The  resistance  of 
the  dielectric  is  to  be  considered  infinite.     The  diameter  of  a  No. 
00  wire  is  0.365  inch. 

Ans.:  3.87. 

11.  What  is  the  displacement  current  per  mile  of  the  above 
cable  when  the  difference  of  potential  between  wire  and  sheath 
varies  at  the  rate  of  10,000  volts  per  second? 

Ans.:  1810  microamperes. 


VI 
VARIABLE  CURRENTS 

158.  Total    Energy    Associated    with    an    Electric    Circuit.  — 

From  the  foregoing  discussion  of  the  magnetic  and  electrostatic 
field  it  is  evident  that  the  energy  associated  with  an  electric 
circuit  may  manifest  itself  in  the  following  ways: 

1.  As  magnetic  energy,  due  to  the  magnetic  field  produced 
by  the  current  in  the  circuit. 

2.  As  electrostatic  energy,  due  to  the  electrostatic  field  pro- 
duced by  the  differences  of  electric  potential  between  the  various 
parts  of  the  circuit. 

3.  As  mechanical  energy,  due  to  the  relative  motion  of  the 
various  parts  of  the  circuit  or  to  the  motion  of  conductors  or  of 
magnetic  or  dielectric  bodies  in  the  surrounding  field. 

4.  As  chemical  energy,  due  to  chemical  changes  which  may 
take  place  in  various  parts  of  the  circuit. 

5.  As  reversible  heat  energy,   due  to  thermal  electromotive 
forces. 

6.  As   heat   energy   dissipated  in  the   conductors  (and  to   a 
slight  extent  in  the  surrounding  dielectric),  due  to  their  electric 
resistance. 

7.  As  heat  energy  dissipated  in  surrounding  magnetic  bodies 
due  to  the  variation  of  the  magnetic  flux  through  them ;  i.e., 
magnetic  hysteresis. 

8.  As   heat   energy   dissipated   in   the  surrounding   dielectric 
bodies  due  to  the  variation  of  the  electrostatic  flux  in  them; 
i.e.,  dielectric  hysteresis. 

The  effects  produced  by  the  first  five  forms  of  energy  are  all 
reversible ;  i.e.,  energy  is  required  to  establish  a  magnetic  or  an 
electrostatic  field,  but  the  same  amount  of  energy  is  given  back 
in  some  other  form  when  these  fields  disappear ;  energy  is  required 
to  move  a  conductor  or  a  magnetic  or  a  dielectric  body  in  the  field, 
but  the  same  amount  of  energy  is  given  back  in  some  other  form 
when  the  motion  of  the  body  is  reversed ;  energy  is  required  to 
produce  any  chemical  action  but  the  same  amount  of  energy  is 

284 


VARIABLE    CURRENTS  285 

given  back  in  some  other  form  when  this  action  is  reversed;  the 
heat  energy  given  out  at  a  junction  of  two  dissimilar  substances 
when  the  current  flows  through  the  junction  in  one  direction,  is 
given  back  as  some  other  form  of  energy  when  the  current  is 
reversed. 

The  effects  produced  by  the  last  three  forms  of  energy  are 
not  reversible;  i.e.,  energy  is  always  dissipated  in  a  conductor 
through  which  a  current  is  flowing,  independent  of  the  direction 
of  the  current ;  energy  is  always  dissipated  when  the  phenomenon 
of  magnetic  or  dielectric  hysteresis  occurs,  independent  of  whether 
the  flux  is  increasing  or  decreasing. 

The  first  five  effects  are  sometimes  said  to  be  due  to  conserv- 
ative forces,  the  last  three  to  dissipative  forces.  The  word  forces 
is  here  used  in  a  general  sense,  meaning  the  something  that  pro- 
duces or  opposes  the  effects. 

159.  General  Equations  of  the  Simple  Electric  Circuit.  — 
The  transfer  of  energy  from  one  region  to  another  in  space  by 
means  of  an  electric  current  is  in  general  accompanied  by  pro- 
duction of  energy  in  all  these  various  forms,  except  in  the  special 
case  of  a  continuous  current,  i.e.,  a  current  which  does  not  vary 
with  time;  there  is  also  in  general  a  transfer  of  energy  from  one 
form  to  another  all  along  the  circuit.  To  understand  clearly  the 
effects  produced  by  a  variable  current  it  is  necessary  to  confine 
one's  attention  at  first  to  comparatively  simple  circuits. 

Consider  a  circuit  consisting  of  a  coil  having  a  resistance  r 
and  an  inductance  L  in  series  with  a  condenser  having  a  capacity 
C  and  a  conductance  g.  The  conductance  of  a  condenser,  that 


Fig.  82. 

is,  the  reciprocal  of  the  resistance  of  the  dielectric  between  its 
plates,  is  sometimes  called  the  leakance  of  the  condenser.  This 
circuit  is  typical  of  the  circuits  one  has  to  deal  with  in  practice; 


286  ELECTRICAL   ENGINEERING 

in  fact  the  most  general  form  of  circuit  may  be  looked  upon  as 
made  up  of  such  elementary  circuits.  This  simple  circuit  may 
be  represented  diagrammatically  as  shown  in  Fig.  82,  the  resist- 
ance and  inductance  and  the  leakance  and  capacity  being  shown 
separately  simply  for  convenience.  Let  an  electromotive  force 
e  be  impressed  across  the  terminals  _of  this  circuit.  This  im- 
pressed electromotive  force  will  establish  a  current  in  the  circuit 
and  therefore  a  potential  drop  across  the  various  parts  of  the 
circuit.  When  there  is  originally  no  current  in  the  circuit  and 
no  charge  on  the  condenser,  this  current  will  be  around  the  circuit 
in  the  direction  of  the  impressed  electromotive  force,  and  will 
produce  a  drop  of  potential  through  the  coil  and  through  the 
condenser.  The  current  in  the  coil  will  have  the  same  value  at 
each  part  of  its  winding  at  any  given  instant  (provided  the  capacity 
and  leakance  of  the  coil  may  be  neglected)  and  this  current  must 
in  turn  be  equal  to  the  total  current  through  the  condenser. 
Let  i  be  the  current  in  the  coil,  id  the  displacement  current  through 
the  condenser  and  ic  the  conduction  current  through  the  con- 
denser. Then  from  Article  149 


dq  dq 

but  id=~7f,  where  -r  represents  the  time  rate  of  change  of  the 

charge  on  the  condenser.  Let  v  be  the  potential  drop  through  the 
condenser  in  the  direction  of  the  current  through  it,  then  from 
Article  149 


The  conduction  current  through  the  condenser  is,  by  Ohm's  Law, 
ic  =gv.     Hence 


From  Ohm's  Law,  the  drop  of  potential  through  the  coil 
due  to  its  resistance  is  ri.  From  Article  116  the  drop  of  poten- 
tial through  the  coil  due  to  its  inductance  (i.e.,  the  back  electro- 

di 

motive  force  of  induction)  is  L  —  .     Hence  the  total  drop  of  po- 

dt 

tential  through  the  coil  is 

.     Tdi 

n+L  — 
dt 


VARIABLE  CURRENTS  287 

and  therefore  the  total  drop  of  potential  through  the  coil  and  the 
condenser  in  series  is 

.     Tdi 

n+L — \-v 
dt 

and  this  must  be  equal  to  the  impressed  electromotive  force  e, 

that  is, 

e  =  ri+L — \-v  (2) 

dt 

Equations  (1)  and  (2)  are  two  equations  in  the  two  unknown 
quantities  i  and  v.  The  solution  of  practically  every  electrical 
engineering  problem  that  has  to  do  with  the  transmission  of 
energy  by  variable  or  alternating  currents  is  simply  a  special 
solution  of  one  or  more  sets  of  equations  of  this  kind.  It  is 
therefore  necessary  to  understand  thoroughly  the  physical  mean- 
ing of  every  term  in  these  two  equations  and  to  become  thoroughly 
familiar  with  the  various  mathematical  expedients  employed 
in  their  solution. 

Since  energy  is  transferred  by  the  current  to  the  electrostatic 
field  of  the  condenser,  the  potential  drop  v  across  the  condenser 
may  be  looked  upon  as  the  back  electromotive  force  of  the  con- 
denser (see  Article  156). 

These  two  equations  may  then  be  looked  upon  simply  as  a 
special  case  of  KirchhofPs  two  laws.  In  the  first  equation  i 
represents  the  current  flowing  to  the  junction  between  the  coil 

and  condenser  and  gv+  C —  represents  the  total  current  flowing 

dt 

away  from  this  junction.     In  the  second  equation,  which  may 

be  written  e  —  v—L—  =  ir,  the  symbols  e,  v,  and  L—  each  repre- 
dt  dt 

sent  an  e.  m.  /.,  the  last  two  being  in  the  opposite  direction  to  the 

impressed  electromotive  force  e,  and  therefore  e  —  v—L—  repre- 

dt 

sents  the  sum  of  the  e.  m.  f.'s  in  the  loop  formed  by  the  source  of 
the  impressed  e.  m.  /.,  the  inductance  L,  the  resistance  r,  and  the 
capacity  C,  and  ir  is  the  total  resistance  drop  in  this  loop.  In 
fact,  Kirchhoff's  Laws  apply  not  only  to  steady  currents  but  to 
currents  varying  in  any  manner  whatever,  provided  the  instan- 
taneous values  of  the  currents  are  considered. 

It  should  be  clearly  borne  in  mind  that  equations  (1)  and  (2) 


288 


ELECTRICAL  ENGINEERING 


apply  directly  only  to  a  single  circuit  and  when  there  is  no  dissipa- 
tion of  energy  due  to  either  magnetic  or  dielectric  hysteresis  or  to 
currents  induced  in  surrounding  bodies.  When  there  is  any 
dissipation  of  energy  due  to  hysteresis  or  eddy  currents*  a  term 
should  be  added  to  take  into  account  this  dissipation  of  energy. 
In  case  there  is  iron  or  any  other  magnetic  substance  in  the 
magnetic  field  produced  by  the  current  there  are  both  magnetic 
hysteresis  and  eddy  currents ;  in  such  a  case  equations  (1)  and  (2) 
give  only  a  first  approximation  to  the  true  relations  between 
current,  potential  difference  and  time;  this  approximation,  how- 
ever, is  usually  sufficiently  close  for  practical  work.  We  shall 
see  in  Article  190  how  a  still  closer  approximation  can  be  made 
in  the  case  of  alternating  currents. 

In  this  chapter  will  be  given  the  solutions  of  equations  (1) 
and  (2)  for  a  few  simple  cases.  In  every  case  the  inductance 
will  be  assumed  constant ;  when  there  is  iron  in  the  magnetic  field 
of  the  current  this  assumption  will  give  only  a  roughly  approxi- 
mate solution. 

160.  Establishment  of  a  Steady  Current  in  a  Circuit  Containing 
T.  Resistance  and  Inductance.  —  The 

circuit  is  represented  diagrammati- 
cally  in  Fig.  83;  E  represents  a 
constant  e.  m.  /.  Let  the  time  t  be 
reckoned  from  the  instant  the 
switch  S  is  closed;  i.e.,  let  the 
switch  be  closed  at  time  t  =0.  At 
any  instant  an  interval  of  time  t 
after  closing  the  circuit. 


MAM/ — OTfr-H 


E 


Pig.  83. 


whence 


Integrating, 


E 


rdf__ 


dt 

d(E-ri)    . 
E-ri 


where  G  is  a  constant  of  integration.     The  value  of  G  is  found  by 

*  The  name    eddy  current  is   applied  to  the  currents    induced   in    the 
magnetic  circuit  or  in  any  metal  forming  the  frame  of  an  electric  machine. 


VARIABLE  CURRENTS  289 

the  condition  that  at  time  Z=0  (that  is,  at  the  instant  the  circuit 
is  closed)  i=0.*  Substituting  these  values  in  the  above  equation 
gives 

G=ln  E 
hence  at  any  instant  t  seconds  after  closing  the  circuit 


L          \    E 

Writing  each  side  of  this  equation  as  the  exponent  of  the  base  e 
of  the  natural  system  of  logarithms,  we  get 
-£_E-ri 

E 
whence 

t-_£  /i_,-r\  (3) 


r 
The  physical  interpretation  of  this  equation  is  that  the  current 

IS 

reaches  its  steady  value  7= —  only  after  the  time  t  measured 

r 

from  the  instant  of  closing  the  circuit  has  become  sufficiently 

great    to    make    the    term   e  L    sensibly  equal  to  zero.     Since, 

r  -- 

however,   the  ratio  —  is  usually  quite  large,  this  term   €   L    as   a 

-L/ 

rule  becomes  practically  zero  for  t  equal  to  a  small  fraction  of 
a  second,   and   therefore   the    current   reaches   its   steady   value 

ET 

7  =  -    almost   immediately   after  the   circuit   is    closed.      When 
the  self-induction  is  large,  as  in  the  case  of  a  coil  wound  on  a 

closed   iron  core,   the   ratio  —  may  be  relatively  small,  in  which 
,  L 

case  several  seconds  may  elapse  before  the  current  reaches  its 
steady  value. 

In  any  case,  after  an  interval  of  time  T=-,  the  current  is 

r 

=0.632  - 


2.71 

That  is,  after  an  interval  of  time  T  =  —,  the  current  reaches  63.2 

r 

*Otherwise  at  the  instant  the  switch  is  closed  energy  would  be  trans- 
ferred to  the  magnetic  field  at  an  infinite  rate,  which  is  impossible. 


290 


ELECTRICAL  ENGINEERING 


per  cent  of  its  final  value.  The  time  required  for  the  current  in 
such  a  circuit  to  reach  100  (1  —  e"1)  =63.2  per  cent  of  its  final  steady 
value  when  a  steady  e.  m.  f.  is  impressed  upon  the  circuit  is  called 
the  "  time-constant  "  of  the  circuit.  The  time-constant  of  such  a 
circuit  may  be  looked  upon  as  a  measure  of  the  slowness  with 
which  the  current  reaches  its  steady  value,  the  greater  the  time- 
constant  the  longer  the  interval  before  the  steady  value  is  reached. 
For  a  circuit  consisting  of  an  inductance  L  and  a  resistance  r  the 

time-constant  is  equal  to  — .     It  should  be  noted  that  this  L  and 

r 

r  refer  to  the  entire  circuit ;  hence  when  the  impressed  e.  m.  f.  is 
produced  by  a  generator  developing  a  steady  armature  e.  m.  f. 
E,  the  resistance  and  inductance  of  the  generator  must  also 
be  included.  In  addition,  the  above  formulas  hold  only  in  case 
the  inductance  of  the  entire  circuit  is  a  constant,  and  there  is 
no  energy  dissipated  in  hysteresis,  which  conditions  never  hold 


Fig.  84. 

when  there  is  iron  or  any  other  magnetic  substance  in  the  mag- 
netic field  produced  by  the  current.  In  such  cases,  however, 
the  formulas  may  be  looked  upon  as  a  first  approximation  to  the 
true  relation  between  current  and  time. 

The  relation  between  current  and  time  given  by  equation  (3) 
may  be  represented  graphically  by  plotting  the  current  i  as  the 
ordinates  against  the  time  t  as  abscissas.  The  curve  has  the 
general  shape  shown  in  figure  84;  i.e.,  the  current  rises  rapidly 
at  first  and  becomes  asymptotic  to  the  line  corresponding  to 


VARIABLE  CURRENTS  291 

/  =  _,  which  gives  the  final  steady  value  of  the  current.     The 
r 

abscissa  T  of  the  point  having  the  ordinate  i  =0.632  —  is  equal  to 

the  time-constant  of  the  circuit. 

161.  Decay  of  Current  in  a  Circuit  Containing  Resistance  and 
Inductance.  —  Let   the  circuit  be  r  L 

short-circuited    on    itself    at    any     | \AAAA/V rfflffiG^ — 

instant  when  the  current  has  the 

value  i0,  say;  let  the  time  t  be 

measured  from  this  instant,  i.e.,  at 

time  2=0  let  i  =i0.     The  circuit  is 

represented     diagrammatically    in  Fig.  85. 

Fig.  85.     In  this  case  the  impressed  e.  m.  /.  is  zero,  whence 


dt 
and  therefore 

r  dt      di 

~L       7 
which  integrated  gives 

-rl=lni+G 
L 

where  G  is  a  constant  of  integration.  The  value  of  G  is  found 
from  the  condition  that  at  time  2=0,  .i=i0  which  values  sub- 
stituted in  the  above  equation  give  G  =  —ln  i.  Hence  at  any 
instant  2  seconds  after  closing  the  circuit 

rt_       i 
~L  =  Ui~0 
whence 


and  therefore 

(4) 


The  physical  interpretation  of  this  equation  is  that  the  current 

does  not  fall  to  zero  immediately,  but  only  after  a  sufficient  time 

jt 

t  has  elapsed  to  make  the  term  iL  sensibly  zero,  which  is  usually 
only  a  fraction  of  a  second,  unless  the  self-induction  of  the  circuit 


292 


ELECTRICAL  ENGINEERING 


is  large  compared  with  its  resistance.    The  current  falls  to  100  e"1  = 

36.8  per  cent  of  its  original  value  in  the  time  T  =— =the  time- 

r 

constant  of  the  circuit.     The  relation  between  the  current  i  and 


Fig.  86. 

time  t  is  shown  graphically  in  Fig.  86.  The  current  falls  rapidly 
at  first  and  becomes  asymptotic  to  the  axis  of  t,  i.e.,  to  the  line 
corresponding  to  i=Q. 

162.  Charging  a  Condenser  through  a  Resistance. — The  cir- 
cuit is  represented  diagrammati- 
cally  in  Fig.  87;  E  is  a  constant 
e.  m.  /.  Let  the  charge  on  the  con- 
denser be  zero  at  the  instant  the 
switch  is  closed,  and  let  time  be 
measured  from  this  instant;  i.e., 
at  time  £=0  let  q0  =0  and  i=Q. 
Let  q  and  i  be  the  charge  and 
current  respectively  and  v  the  p.d. 
through  the  condenser  at  any  in- 
stant t  seconds  after  closing  the  switch.  Then  at  this  instant, 
assuming  the  ideal  conditions  of  no  leakance  in  the  condenser  and 
no  inductance  of  the  circuit,  -,_  . 


— MAM 


E 


Fig.  87. 


dt 


E-v 


From  the  first  equation  we  have  that  i  = 
in  the  second  equation  gives  r 


which  substituted 


VARIABLE  CURRENTS  293 


dt__     dv        -d(E-v) 
rC~E-v~       E-v 
which  integrated  gives 


where  G  is  a  constant  of  integration.  For  2=0,  v=0,*  which 
substituted  in  this  equation  gives  G=  —  In  E.  Substituting  this 
value  of  G  in  the  last  equation  gives  the  equation 

-L-Jn  (* 
rC          \    E 

whence 


(5a) 

which  substituted  in  the  second  equation  above  gives 
,_E    -± 

r 
and  since  q  =  Cv  at  any  instant  we  get  immediately  from  (5a)  that 

q  =  C  E   (l-  i'^c 


Fig    88. 


Equations  (5)  then  give  the  values  of  the  p.d.  through  the  con- 
denser, the  current,  and  the  charge  at  any  instant.  The  variation 
of  v,  it  and  q  with  time  is  shown  graphically  in  Fig.  88.  Note 
that  both  the  p.d.  through  the  condenser  and  the  charge  start  at 

*Otherwise  at  the  instant  the  switch  is  closed  energy  would  be  transferred 
to  the  electrostatic  field  at  an  infinite  rate,  which  is  impossible. 


294 


ELECTRICAL  ENGINEERING 


zero  rise  rapidly  at  first  and  then  approach  more  slowly  their 
constant  values  E  and  CE  respectively.  The  current,  on  the 

Tjl 

other  hand,  has  its  maximum  value  -  at  the  instant  the  switch 

r 

is  closed  and  then  falls,  first  rapidly  and  then  more  slowly,  to  zero.* 
As  a  rule  rC  is  extremely  small,  so  that  only  a  small  fraction  of 
a  second  is  required  for  these  steady  conditions  to  become 
established. 

The  product  rC  measures  the  slowness  with  which  the  con- 
denser becomes  charged;  it  is  called  the  time-constant  of  the 
circuit.  Compare  with  the  time-constant  of  a  circuit  formed  by 
a  resistance  and  an  inductance  in  series. 

163.  Discharging  a  Condenser 
through  a  Resistance.  —  Let  the  con- 
denser be  charged  to  a  p.d.v0  at 
the  instant  t  —  0  at  which  it  is  short- 
circuited  through  a  resistance  r,  Fig. 
89.  Assuming  as  before  the  ideal  con- 
ditions of  no  leakance  and  no  indue- 


MAM 


Fig.  89. 

tance,  we  have 


dt 
Eliminating  i  from  these  two  equations  gives 


whence 


whence 


n  av 

-rC  —  =v 
dt 


dt      dv 


rC 

But  at  t  =0,  v  =v0,  whence  G  =  —  Inv0. 

Hence  at  any  instant  t  seconds   after  the  condenser  is  short- 

circuited, 

*Actually  the  current  always  starts  from  zero  and  rises  to  a  maximum 
value,  due  to  the  inductance  of  the  circuit,  which,  though  it  may  be  small, 
is  never  absolutely  zero  as  assumed  in  the  above  discussion;  see  Article  201. 


VARIABLE  CURRENTS 


295 


V=V0€   rC 


!^€ 

r 


fc" 


j_ 

rC 


(6o) 

(66) 

(6c) 


In  this  case  both  the  p.d. 
and  the  charge  decrease  at 
first  rapidly  and  then  more 
slowly  to  zero.  The  current 
through  the  condenser,  how- 
ever, is  negative,  i.e.,  in  the 
opposite  direction  to  the  drop 
of  potential  through  the  con- 
denser. Hence  the  current 
through  the  resistance  is  in 
the  direction  from  the  +  to 
the  —  plate  of  the  condenser. 
The  current  is  a  maximum 

—  (in  the  negative  direction) 
r 

at   the   instant    of    short    cir- 

cuit,*   and  then  decreases  in  the  same  manner  as  the  p.d.  and 

the  charge. 

164.  Discharge  of  a  Condenser  through  an  Inductance.  —  The 
circuit  is  represented  diagrammatically  in  Fig.  91.  The  condenser 
is  charged  to  a  p.d.v0  and  at  time  t  =  0  the  switch  is  closed.  At 
any  time  t  seconds  later  assuming  the  ideal  conditions  of  no  leakance 
and  no  resistance, 


Fig 


dt 


(a) 


Fig.  91. 

equation  gives 


From     equation    (6)    we    have    that 

^/?  (i    i) 

—  =  C  —  which  substituted  in  the  first 
dt       dt2 


*  The  current  actually  starts  at  zero  due  to  the  inductance  of  the  cir- 
cuit; see  Article  201. 


296  ELECTRICAL  ENGINEERING 

tfv  1 

= V  (c) 

dt2  LC 

This  equation  is  of  exactly  the  same  form  as  the  equation  for 
'the  pendulum;  see  Article  23.  Compare  also  with  the  equation 
for  the  vibration  of  a  magnet  in  a  uniform  magnetic  field,  Article  42. 
Its  solution  is 


which  substituted  in  equation  (6)  gives 

C         f 


L      '  VVLC 

The  constants  A  and  6  are  determined  from  the  condition  that 
at  time  t=Q,  v=v0,  and  i=0.  Substituting  these  values  in  (d) 
and  (e)  we  get 

vn  =  A  sin  0 


Whence  0=—  and  A  =v0.     The  equations  for  the  p.d,  arid  cur- 
2 

rent  at  any  instant  are  then 


(76) 

These  two  equations  are  plotted  in  Fig.  92.  That  is,  both  v  and  i 
are  harmonic  functions,  or  "  sine  waves/'  having  a  period  equal 
to  2  TT  \/LC.  The  maximum  value  of  the  p.d.  is  v0  and  the  maxi- 

fc 

mum  value  of  the  current  is  I>OA|  —  That  is,  the  charge  on  the 
condenser  oscillates  from  q  =  Cv0  to  —  q  —  —  Cv0,  and  the  current 

|c  |c 

oscillates   between   the  values  v0  \l  -  and  —  v0^l  -,  the  charge 

'  L  '  L 

reaching  the  maximum  when  the  current  is  zero  and  vice  versa. 

If  the  current  in  the  inductance  had  been  equal  to  i0  and  the 
p.d.  across  the  condenser  zero  at  time  £=0,  then  the  equations 
for  p.d.  and  current  would  have  been 


VARIABLE  CURRENTS 


297 
(8a) 
(86) 


Fig.   92. 


A  circuit  of  this  kind  is  approximately  realized  in  the  case  of 
a  transmission  line,  except  that  the  inductance  and  capacity  in- 
stead of  being  lumped  in  a  single  coil  and  a  single  condenser  are 


Fig.  93. 


distributed  uniformly  along  the  line,  and  the  resistance  of  the  line 
is  not  negligible.  As  a  first  approximation  we  may  consider  the 
line  as  equivalent  to  a  lumped  inductance  and  a  lumped  capacity 
as  shown  in  Fig.  93  and  neglect  the  resistance.  If  the  line,  open 


298  ELECTRICAL  ENGINEERING 

at  the  receiving  end,  becomes  short-circuited  at  any  time  P,  by 
a  wire  falling  across  it,  for  example,  we  have  the  conditions  just 
discussed. 

Take  the  case  of  a  line  consisting  of  two  number  0000  B.  &  S. 
wires  (solid)  spaced  6  feet  apart;  let  the  line  be  50  miles  long 
(100  miles  of  wire). 

The  inductance  of  the  line  will  then  be  0.193  henries  and  the 
capacity  0.389  x  10"6  farads.  Let  500  amperes  be  flowing  in  the 
line  at  the  instant  of  short-circuit  at  P  and  let  the  p.d.  across 
the  end  of  the  line  (i.e.,  across  the  condenser)  be  zero  at  this 
instant.  Then  this  p.d.  will  reach  a  maximum  of 

500  J Q'193     =352,000  volts 

^  0.389  X 10  -6 

and  the  current  and  p.d.  will  begin  to  oscillate  with  a  period  of 


2  ?r\/0.193X0.389XlO-6=0.00172  seconds  or  580  com- 
plete swings  or  cycles  per  second.  This  abnormal  rise  in  voltage 
would  of  course  puncture  the  insulators  carrying  the  wires  were 
not  some  sort  of  "  safety  valve  "  provided.  Such  a  safety  device 
is  the  lightning  arrester,  which  in  its  simplest  form  consists  of  one 
or  more  spark  gaps  so  arranged  that  when  the  voltage  on  the  line 
rises  a  predetermined  amount  above  normal,  a  spark  jumps  across 
the  gap  and  thus  reduces  the  voltage.  Protecting  a  line  against 
lightning  discharges  is  only  one  of  the  functions  of  a  lightning 
arrester;  such  a  device  is  likewise  necessary  to  protect  the  line 
(and  the  apparatus  connected  therewith)  against  the  abnormal 
voltage  which  may  be  produced  when  heavy  loads  are  switched 
on  or  off  the  line. 

Note  that  the  ideal  case  of  a  condenser  without  leakance  and 
an  inductance  without  resistance,  which  we  have  been  consider- 
ing, cannot  be  completely  realized  in  practice,  since  the  dielectric 
of  every  condenser  is  a  conductor  to  a  certain  extent  (though 
usually  an  extremely  poor  conductor)  and  every  circuit  made  of 
conductors  has  a  certain  resistance.  The  effect  of  resistance  and 
leakance  is  to  damp  out  the  oscillations  of  the  electric  current  in 
the  same  way  that  friction  damps  out  the  oscillation  of  a  vibrating 
pendulum.  (See  Article  202.) 


VARIABLE  CURRENTS  299 

SUMMARY  OF  IMPORTANT  RELATIONS 

1.  In  general,  the  energy  associated  with  an  electric  current 

may  manifest  itself  in  eight  different  forms,  i.e.,  as 
Magnetic  energy, 
Electrostatic  energy, 
Mechanical  energy, 
Chemical  energy, 

Heat  energy  at  the  junction  of  dissimilar  substances, 
Heat  energy  due  to  the  resistance  of  the  conductors, 
Heat  energy  due  to  magnetic  hysteresis, 
Heat  energy  due  to  dielectric  hysteresis. 

2.  The  general  equations  of  a  circuit  formed  by  a  coil  and  a 
condenser  in  series  are 

i    ndv 

i  =  gv+C- 
at 

di 

—  +  v 
dt 

where  e  is  the  impressed  e.  m.  f.  across  the  terminals  of  the  circuit, 
v  the  potential  drop  through  the  condenser,  i  the  current  in  the 
coil,  r  the  resistance  of  the  coil,  L  the  inductance  of  the  coil,  g 
the  leakance  of  the  condenser  and  C  the  capacity  of  the  condenser. 
These  two  equations  are  simply  KirchhofPs  Law  applied  to  the 
instantaneous  values  of  the  current,  e.  m.  /.  and  p.  d. 

3.  The  current  i  in  a  circuit  formed  by  a  resistance  r  and  an 
inductance  L,  t  seconds  after  impressing  a  constant  e.  m.  f.   E 
across  its  terminals,  is 


provided  there  is  originally  no  current  in  the  circuit;  r  is  the 
resistance,  L  the  inductance  of  the  entire  circuit. 

4.  The  time-constant  of  a  circuit  formed  by  a  resistance  r  and 

an  inductance  L  in  series  is  — . 

r 

5.  The  current  i  in  a  circuit  formed  by  a  resistance  r  and  an 
inductance  L,  t  seconds  after  the  terminals  of  the  circuit  are  short- 
circuited,  is  rt 

i  =  i0  €  L 

where  i0  is  the  current  in  the  circuit  at  the  instant  of  short-circuit, 
provided  there  is  no  e.  m.  f.  in  the  circuit. 


300  ELECTRICAL  ENGINEERING 

6.  When  a  condenser  of  capacity  C  is  charged  through  a 
resistance  r  by  a  constant  impressed  e.  m.  /.  E,  the  p.  d.  across  the 
condenser,  the  current  in  the  circuit  and  the  charge  on  the  con- 
denser t  seconds  after  the  e.  m.  /.  is  impressed,  are  respectively 


V  =  E  (l  -  €~rC\ 


provided  there  is  originally  no  charge  on  the  condenser  and  no 
current  in  the  circuit  and  there  is  no  inductance  in  any  part  of 
the  circuit. 

7.  The  time-constant  of  a  circuit  formed  by  a  resistance  r  and 
a  capacity  C  in  series  is  rC. 

8.  When  a  condenser  of  capacity  C,  through  which  the  p.d. 
is  v0.  is  discharged  through  a  resistance  r,  the  p.d.  through  the 
condenser,  the  current  in  the  circuit  and  the  charge  on  the  con- 
denser t  seconds  after  the  terminals  of  the  circuit  are  short-circuited 

are  respectively 

_j_ 
v=v0  e  rC 


q  =  Cv0  €  rC 

provided  there  is  no  current  in  the  circuit  at  the  instant  of  short- 
circuit  and  there  is  no  e.  m.  /.  and  no  inductance  in  the  circuit. 

9.  When  a  condenser  of  capacity  C,  through  which  the  p.d.  is 
v0,  is  discharged  through  an  inductance  L,  the  p.d.  through  the 
condenser  and  the  current  in  the  circuit  t  seconds  after  the  ter- 
minals of  the  circuit  are  short-circuited  are  respectively 

t 


provided  there  is  no  current  in  the  circuit  at  the  instant  of  short- 
circuit  and  there  is  no  e.  m.  f.  and  no  resistance  in  the  circuit. 
When  the  terminals  of  the  circuit  are  short-circuited  at  the  instant 


VARIABLE  CURRENTS  301 

when  the  current  is  i0  and  there  is  no  p.d.  through  the  condenser, 
the  p.d.  and  current  t  seconds  after  the  terminals  are  short-circuited 
are  respectively 


ln  COS 


( 


WWJ 

provided  there  is  no  e.  m.  f.  and  no  resistance  in  the  circuit. 

PROBLEMS 

1.  An  air-core  solenoid  of  5000  turns  is  20  cm.  long  and  has  a 
diameter  of  5  cm. ;  the  resistance  of  the  solenoid  is  2  ohms.     When 
a  constant  e.  m.  /.  of  50  volts  is  impressed  across  the  terminals  of 
this  coil  (1),  at  what  rate  does  the  current  begin  to  increase;  (2) 
what  is  the  final  value  of  the  current;  and  (3)  what  is  the  time- 
constant  of  the  coil? 

Ans.:  (1)  162.2  amperes  per  second;  (2)  25  amperes;  (3)  0.154 
second. 

2.  In  the  preceding  problem,  (1)  at  what  rate  is  energy  being 
stored  in  the  magnetic  field  of  the  current  when  the  current  has 
reached  a  value  of  10  amperes ;  (2)  when  the  energy  of  the  electro- 
magnetic field  is  30.8  joules  at  what  rate  is  energy  dissipated  in 
heating  the  coil ;  (3)  at  what  rate  is  the  current  changing  when  the 
total  power  supplied  to  the  coil  is  250  watts? 

Ans.:  (1)   300  watts;  (2)   400  watts;  (3)    129.9  amperes  per 
second. 

3.  At  the  same  instant  that  the  e.  m.  f.  impressed  upon  a  coil 
is  removed,  the  ends  of  the  coil  are  connected  by  a  resistance  of 
3  ohms.     At  the  instant  immediately  after  this  change  in  con- 
nections is  made  the  current  in  the  coil  is  12  amperes  and  is  de- 
creasing at  a  rate  of  480  amperes  per  second.     (1)  What   is  the 
time-constant  of  the  circuit?      (2)  If  the  energy  of  the  electro- 
magnetic field  is  7.2   joules  at  the  instant  that  the   change  in 
connections  is  made,  at  what  rate  is   energy   dissipated  in  the 
circuit  in  the  form  of  heat  energy  at  that  instant?    (3)  What  is 
the  resistance  and  the  inductance  of  the  coil? 

Ans.:  (1)  0.025  second;  (2)  576  watts;  (3)  resistance  1  ohm, 
inductance  0.1  henry. 

4.  An  e.  m.  f.  of  20  volts  is  impressed  upon  a  coil  of  0.6  ohm 


302  ELECTRICAL  ENGINEERING 

resistance  and  0.3  henry  inductance.     What  is  the  value  of  the 
current,  in  the  coil  0.5  second  after  the  e.  m.  /.  is  impressed? 
Ans.:  21.1  amperes. 

5.  An  e.  m.  f.  of  30  volts  is  impressed  upon  a  coil  of  0.5  ohm 
resistance  and  0.2  henry  inductance.     One  tenth  of  a  second  after 
the  e.  m.  /.  is  impressed,  the  source  of  the  e.  m.  f.  is  removed  and 
the  coil  is  short-circuited  by  a  resistance  of  0.1  ohm.     Determine 
the  value  of  the  current  in  the  circuit  0.08  second  later. 

Ans.:  10.48  amperes. 

6.  A  battery  which  has  an  e.  m.  f.  of  10  volts  and  a  resistance 
of  0.5  ohm,  a  coil  which  has  an  inductance  of  3  henries  and  a 
resistance  of  0.5  ohm,  and  a  non-inductive  resistance  of  4  ohms 
are  all  connected  in  series  to  form  a  closed  electric  circuit.     Find 
the  value  of  the  current  flowing  in  a  non-inductive  resistance  of 
4  of  an  ohm  2  seconds  after  it  is  connected  in  parallel  with  the 
4  ohm  resistance. 

Ans.:  4.34  amperes. 

7.  An  e.  m.  f.  of  250  volts  is  impressed  upon  a  circuit  con- 
formed by  a  non-inductive  resistance  of  1000  ohms  in  series  with 
a  condenser   of   50   microfarads  capacity.      Find  (1)  the  initial 
rate  at  which  the  condenser  is  charged;  (2)  the  current  in  the 
circuit  when  the  charge  on  the  condenser  is  5000  microcoulombs ; 

(3)  the  energy  in  the  electrostatic  field  when  energy  is  dissipated 
as  heat  energy  in  the  circuit  at  the  rate  of  10  watts;  and  (4)  the 
charge  in  microcoulombs  on  the  condenser  when  the  current  is 
changing  at  the  rate  of  4  amperes  per  second. 

Ans.:  (1)  0.25   ampere;  (2)    0.15   ampere;  (3)    0.563   joule; 

(4)  2500  microcoulombs. 

8.  A  100  microfarad  condenser  charged  to  a  potential  differ- 
ence of  600  volts  and  is  discharged  through  a  resistance  of  500 
ohms.     Find  (1)  the  current  when  the  current  is  decreasing  at 
the  rate  of  16  amperes  per  second;  (2)  the  charge  on  the  con- 
denser when  the  energy  of  the  electrostatic  field  is  2  joules;  (3) 
the  potential  drop  through  the  condenser  when  the  condenser  is 
discharging  at  the  rate  of  0.5  coulomb  per  second;  and  (4)  the  rate 
at  which  the  condenser  is  losing  energy  when  the  charge  on  the 
condenser  is  10,000  microcoulombs. 

Ans.:  (1)  0.8  ampere;  (2)  20,000  microcoulombs ;  (3)  250  volts ; 
(4)  20  watts. 

9.  One-half  a  second  after  an  e.  m.  f.   of  1000  volts   is  im- 


VARIABLE  CURRENTS  303 

pressed  upon  a  circuit  formed  by  a  non-inductive  resistance  and  a 
condenser  in  series,  the  current  is  10  rnilliamperes.     If  the  resist- 
ance is  10,000  ohms,  what  is  the  capacity  of  the  condenser? 
Ans.:  21.7  microfarads. 

10.  An  electrostatic  voltmeter  connected  across  the  terminals 
of  a  condenser  indicates  a  potential  difference  of  500  volts.     The 
capacity  of  the  condenser  is  35  microfarads  and  that  of  the  volt- 
meter is  5  microfarads.     When  an  unknown  resistance  is  connected 
across  the  condenser  terminals,  it  is  noted  that  when  the  con- 
denser has  been  discharging  for  1  second  the  voltage  across  its 
terminals  has  decreased  to  100  volts.     What  is  the  value  of  the 
unknown  resistance? 

Ans.:  15,530  ohms. 

11.  Two-tenths  of  a  second  after  an  e.m.f.  of  100  volts  is 
impressed  across  the  terminals  of  a  circuit  formed  by  a  non- 
inductive  resistance  of  5000  ohms  in  series  with  a  20  microfarad 
condenser,  a  30  microfarad  condenser  is  connected  in  parallel  with 
the  20  microfarad  condenser.     Find  the  charge  on  the  30  micro- 
farad condenser  0.3  second  after  this  parallel  connection  is  made. 

Ans.:  2410  microcoulombs. 

12.  A  50  microfarad  condenser  is  charged  to  a  potential  differ- 
ence of  1000  volts  and  is  then  discharged  through  a  coil  of  0.3 
henry  inductance  and  negligible  resistance.     Find  (1)  the  maxi- 
mum value  of  the  current  in  the  circuit;  and  (2)  the  frequency  at 
which  it  oscillates.     (3)  Plot  to  scale  the  current  and  p.d.  waves. 

Ans.:  (1)  12.91  amperes;  (2)  41.1  cycles  per  second. 


VII 


ALTERNATING  CURRENTS 

165.  Introduction.  —  We  have  just  seen  that  when  a  con- 
denser is  discharged  through  a  coil  having  an  inductance  L  but 
no  resistance  that  the  current  in  the  circuit  and  the  p.d.  across 
the  condenser  vary  harmonically,  i.e.,  the  instantaneous  value 
of  the  current  at  any  instant  is  i=I0sin  cut  and  the  instantaneous 
value  of  the  p.d.  across  the  condenser  at  any  instant  is  v  =  V0cos  a)t 
where  I0  and  V0  are  the  maximum  values  of  the  current  and  p.d. 
and  a)  is  equal  to  the  factor  2  TT  divided  by  the  period  of  oscil- 
lation of  the  current  and  p.d.     Both  the  current  and  p.d.  vary 
periodically  from  fixed  maximum  values  (I0  and  V0  respectively) 
in  one  direction  to  equal  maximum  values  ( — 10  and  —  V0  respec- 
tively) in  the  other  direction  and  back  again  to  the  maximum 
values  in  the  first  direction.     The  current  is  therefore  called  an 
alternating  current  and  the  p.d.  an  alternating  p.d.     This  ideal 
circuit  we  have  been  considering  is  a  very  special  one;  alternat- 
ing  currents  employed  in  practical  work   are  produced  by  an 
entirely  different  method,  namely,  by  the  rotation  of  one  or  more 
coils  in  a  magnetic  field.     The  machine  for  producing  alternating 
currents  in  this  manner  is  called  an  alternator. 

166.  The  Simple  Alternator.  —  The  simplest  form  of  alternator 
consists  of  a  coil  of  wire  of  one  turn  rotating  in  a  uniform  magnetic 


Fig.  94.- 

field,  see  Fig.  94.     The  ends  of  the  wire  forming  the  coil  are 
connected  to  two  rings  called  slip  rings;  making  contact  with 

304 


ALTERNATING  CURRENTS  305 

these  rings  are  the  brushes  1  and  2.  Let  the  coil  be  rotated  with 
an  angular  velocity  o> ;  let  time  be  counted  from  the  instant  when 
the  wire  a  is  above  b  and  the  plane  of  the  coil  coincides  with  a 
vertical  plane  drawn  through  the  axis  of  rotation,  and  let  <f>0  be 
the  value  of  the  magnetic  flux  of  induction  threading  the  coil 
when  in  this  position.  Then  <f>  =<j>0  cos  a>t  is  the  value  of  the  flux 
threading  the  coil  at  any  instant  t;  hence  the  e.  m.  f.  induced  in 
the  coil  at  this  instant  is;  see  Article  105, 

d<&         , 

e  =  —  — £•  =ft>  <f>0  sin  CD  t 
dt 

and  is  in  the  direction  from  1  to  2.     The  maximum  value  of  the 

/7T  TT 

e.  m.  /.  is  then  a)  <f>0  and  occurs  when  o>  t  =—  or  t  —  — .     Calling  E0 

2<y 

this  maximum  value  of  the  e.  m.  /.,  the  induced  e.  m.f.  in  the  direc- 
tion from  1  to  2  may  be  written 

e  —  E0  sin  a)  t 
The  electromotive  force  in  this  case  is  a  harmonic  or  sine  function 

of  the  time,  the  period  is   T  =  —  and  the  number  of  complete 

6) 

periods  per  second,  or  the  frequency,  is  /  = — . 

2ir 

In  alternators  as  actually  constructed  the  coils  of  wire  are 
not  made  of  a  single  turn,  nor  do  they  rotate  in  uniform  magnetic 
fields ;  the  field  of  the  alternator  also  has  a  large  number  of  poles. 
By  properly  designing  the  coils  and  the  pole  faces  of  such  a  ma- 
chine it  is  however  possible  to  produce  in  the  coils  when  rotated 
at  constant  speed  an  alternating  e.  m.  /.  which  is  practically  a 
harmonic  function  of  the  time.  In  any  case,  the  e.  m.  /.  will  be 
a  periodic  function  of  the  time  and  can  therefore  be  represented 
by  a  series  of  harmonic  terms,  called  a  Fourier's  Series,  of  the 
form 

e  =  Elsin  a)  t+E2sin  (2 a) t  +  02)  +  E3sin  (3a)t  +  03)  +  - 
where  a)  depends  solely  upon  the  number  of  pairs  of  field  poles 
and  the  angular  velocity  of  the  rotating  part  of  the  machine 
(which  rotating  part  may  be  either  the  field  or  the  armature)  and 
the  E's  and  0's  are  constants,  in  general  different  for  each  term. 
The  first  term  El  sin  a)  t  is  called  the  fundamental  or  first  harmonic, 
the  remaining  terms  are  called  the  second,  third,  etc.,  harmonics. 
The  frequency  of  the  nth  harmonic  is  equal  to  n  times  the  fre- 


306  ELECTRICAL  ENGINEERING 

quency  of  the  fundamental.  In  most  practical  cases  it  is  neces- 
sary to  consider  only  the  first  harmonic  or  fundamental. 

In  order  to  understand  the  properties  of  alternating  electro- 
motive forces  and  currents  it  is  necessary  first  to  get  clearly  in 
mind  certain  fundamental  definitions. 

167.  Definition  of  Alternating  Current  and  Alternating  Electro- 
motive Force.  —  An  alternating  current  is  defined  as  a  current 
which  varies  continuously  with  time  from  a  constant  maximum 
value  in  one  direction  to  an  equal  maximum  value  in  the  opposite 
direction  and  back  again  to  the  same  maximum  in  the  first  direc- 
tion, repeating  this  cycle  of  values  over  and  over  again  in  equal 


Fig.  95. 

intervals  of  time  T,  in  such  a  manner  that  the  instantaneous 
value  of  the  current  at  any  instant  t  is  identically  the  same  as 
at  any  other  instant  t  +  kT  where  T  is  the  time,  constant  in  value, 
required  for  the  current  to  pass  through  a  complete  cycle  of 
values,  and  k  is  any  integer,  positive  or  negative.  \  Similarly,  an 
alternating  e.  m.  f.  is  defined  as  an  e.  m.  /.  which  varies  continu- 
ously with  time  from  a  constant  maximum  value  in  one  direction 
to  an  equal  maximum  value  in  the  opposite  direction  and  back 
again  to  the  same  maximum  in  the  first  direction,  repeating  this 
cycle  over  and  over  again  in  equal  intervals  of  time  T,  in  such  a 
manner  that  the  instantaneous  value  of  the  e.  m.  f.  at  any  instant 
t  is  identically  the  same  as  at  any  other  instant  t  +  kT  where  T  is 
the  time,  constant  in  value,  required  for  the  current  to  pass  through 
a  complete  cycle  of  values,  and  k  is  any  integer,  positive  or  nega- 
tive. In  Fig.  95  the  successive  values  of  an  alternating  current 
are  plotted  as  ordinates  against  time  as  the  abscissa.  Such  a 


ALTERNATING  CURRENTS  307 

curve  is  called  a  current  "  wave."  In  the  figure  the  positive  and 
negative  portions  of  the  current  wave  are  shown  unsymmetrical  ; 
such  a  non-symmetrical  current  or  p.d.  wave  is  physically  possible', 
but  the  current  and  p.d.  waves 
developed  in  ordinary  electric 
machines  are  usually  perfectly 
symmetrical,  i.e.,  the  positive  and 
negative  portions  of  the  wave  are 
exactly  alike. 

An  oscillating  current  or  e.m.f. 
is  a  current  or  e.  m.  /.  which  alter- 
nates in  direction  but  changes  in 
amplitude.  An  oscillating  cur- 
rent wave  is  shown  in  Fig.  96. 

A  pulsating  current  is  a  current  which  varies  with  time  but  is 
always  in  the  same  direction  ;  such  currents  are  obtained  from  an 
arc  rectifier. 

163.  Period,  Frequency,  Alternations,  Periodicity  and  Phase.  —  • 
To  avoid  repetition  the  following  definitions  are  given  in  terms 
of  an  alternating  current;  they  also  apply  to  an  alternating  elec- 
tromotive force,  an  alternating  potential  difference,  or  to  any 
other  periodic  function  of  time. 

The  period  of  an  alternating  current  is  the  time  taken  for 
the  current  to  pass  through  a  complete  cycle  of  positive  and 
negative  values  ;  i.e.,  the  period  is  equal  to  the  time  T  denned  in 
the  preceding  article. 

The  frequency,  or  number  of  cycles  per  second,  is  the  number 
of  periods  per  second. 

The  number  of  alternations  per  minute  is  the  total  number  of 
times  per  minute  that  the  current  changes  in  direction,  from 
positive  to  negative  and  fronl  negative  to  positive.  In  engineer- 
ing practice  the  number  of  cycles  is  usually  referred  to  the  second 
as  the  unit  of  time  and  the  number  of  alternations  is  referred  to 
the  minute  as  the  unit  of  time. 

Let  T  be  the  period,  /  the  frequency  or  number  of  cycles  per 
second,  and  a  the  number  of  alternations  per  minute,  then 


.-120/-™  (2) 


308  ELECTRICAL  ENGINEERING 

The  equation  of  a  harmonic  or  sine-wave  current  is 

i=I0  sin  ((At +  8)  (3) 

where  o>  is  a  constant  equal  to  —   and  8  is  a  constant  such  that 

I0sin  8  gives  the  value  of  the  current  at  time  £=0.  The  constant 
a)  is  called  the  periodicity  of  the  current,  and  the  constant  8  is 
called  the  phase  of  the  current.  The  relation  between  periodicity, 
period  and  frequency  is 

<u=Y=2ir/  (4) 

y  169.  Difference  in  Phase.  —  In  general,  when  a  harmonic  or 
sine-wave  electromotive  force  is  impressed  on  a  circuit  the  result- 
ing current  is  likewise  a  harmonic  function  of  time  (after  a  very 
brief  interval)  having  the  same  frequency,  but  the  e.  m.  /.  and 
current  do  not  reach  their  maximum  values  simultaneously.  In 
other  words,  when  e  =  E0  sin  at  represents  the  e.  m.  /.,  the  current 
is  represented  by  i=I0sin  (ait  +  ff)  where  E0  and  I0  are  the  maxi- 
mum values  of  the  e.m.f.  and  current  respectively;  o>=2  TT  f 
where  /  is  the  frequency ;  t  is  the  time  measured  from  the  instant 
when  e=0  and  is  increasing  in  the  positive  direction  and  8  is  an 
angle  which  measures  the  interval  between  the  instants  when  the 
e.  m.  /.  and  current  reach  successive  maximum  values.  The  e.  m.  /. 

reaches  its  maximum  value  at  time  t  =  —     while  the  current 

2  co, 

reaches  its  maximum  value  when 


2  a)  \2        I 

Hence  when  8  is  positive  the  current  reaches  its  maximum  value 
f\ 

—^seconds   before    the  e.m.f.   reaches    its  maximum ;  when  8  is 

(0  Q 

negative  the  current  reaches  its  maximum  value  —  seconds  after 

O) 

the  e.  m.  f.  reaches  its  maximum.  In  the  first  case  the  current  is 
said  to  lead  the  e.  m.  /.,  and  in  the  second  case  the  current  is  said 
to  lag  behind  the  e.  m.  f.  The  angle  8  is  called  the  difference  in 
phase  between  the  current  and  e.  m.  /.  A  careful  re-reading  of 
the  latter  part  of  Article  23  will  assist  the  reader  in  obtaining  a 
clear  physical  conception  of  phase  difference. 

When  the  phase  difference  is  zero  the  current  and  e.  m.  /.  are 


ALTERNATING  CURRENTS  309 

said  to  be  in  phase;  when  the  phase  difference  is  —  radians  or  90° 

the  current  and  e.  m.  f.  are  said  to  be  in  quadrature;  when  the 
phase  difference  is  TT  radians  or  180°  the  current  and  e.m.f.  are 
said  to  be  in  opposition. 

In  general,   when  we  have  any  two  harmonic  functions  of 

time  and  of  the  same  frequency    -  ,  such  as 

2  7T 

xl  =Xl  sin  ( 

and 

X2=X2  sin 

the  first  function  reaches  its  maximum  value  at  time  t 


ime  tv  =—  (-—  0l  ) 
\2         ' 


and  the  second  reaches  its  first  maximum  value  for  t2  =—  I  —  —  02  }  . 

ft>\2         / 

Hence  the  first  function  xv  reaches  its  first  maximum  an  interval 

of  time  t2—tl  =—  (#i—  #2)  ahead  of  the  second  function  x2;  the  first 
ft) 

function  xl  is  therefore  said  to  lead  the  second  function  x2  by  the 
angle  0!—02.  Note  the  order  of  the  subscripts:  xl  leads  x.2  by 
&i  —  6  2,'  or  x2  leads  x1  by  the  angle  02  —  0lf  A  negative  lead  is 
of  course  equivalent  to  an  actual  lag,  and  a  negative  lag  is  equiv- 
alent to  an  actual  lead. 

170.  Instantaneous,  Maximum,  and  Average  Values.  —  The 
instantaneous  value  of  an  alternating  current  is  the  value  of  the 
current  at  any  instant. 

The  maximum  value  of  an  alternating  current  is  the  greatest 
instantaneous  value  during  any  cycle.  For  a  harmonic  alternat- 
ing current  the  value  of  the  current  at  any  instant  is  given  by  an 
equation  of  the  form  i=I0sin  (a)t  +  0)',  in  this  case  the  constant 
I0  is  the  maximum  value. 

The  average  value  of  an  alternating  current  is  defined  as  the 
numerical  value  of  the  average  of  its  instantaneous  values  between 
successive  zero  values  ;  hence,  when  the  instantaneous  values  are 
plotted  as  ordinates  against  time  as  abscissa,  the  average  value  is 
the  average  ordinate  for  any  positive  half  cycle  of  instantane- 
ous values.  In  the  case  of  a  harmonic  current  of  the  form 
i=I0sin(a)t  +  0)  the  average  value  7aver.  is  equal  to  2  over  TT 
times  the  maximum  value  I0,  that  is, 


310  ELECTRICAL  ENGINEERING 


. 

77 

For,  put  a)t+0  =  x,  then  the  average  value  of  E0  sin  (cot  +  0) 
between  the  limits  CD  t  +  0  =0  and  co  t  +  Q=ir  is  equal  to  the  aver- 
age value  of  I0  sin  x  between  the  limits  x  =0  and  x  =TT,  or 


1    /V  j  p  qTr   2 

~   I    ^o  s^n  2  ^  =—  I  —  cos  x     =—I0 

v  J  o  77-L  J0   TT 


The  average  value  over  a  complete  period  of  an  alternating  current 
having  symmetrical  positive  and  negative  values  is  zero,  since  the 
average  of  the  positive  instantaneous  values  over  half  a  period  is 
equal  to  the  average  of  the  negative  instantaneous  values  over 
half  a  period. 

The  above  definitions  and  deductions  also  apply  to  alternat- 
ing electromotive  forces  and  alternating  potential  differences. 
In  the  case  of  any  harmonic  function  of  the  form  x  =X0  sin  (a*  t  +  0) 
the  same  relation  exists  between  its  maximum  and  average  value 
as  between  the  maximum  and  average  value  of  a  harmonic  cur- 
rent, that  is, 


Xavcr_=-X0  (So) 

TT 


171.  Effective  Values.  —  The  total  amount  of  heat  energy 
developed  during  a  complete  period  T  in  a  resistance  r  through 
which  an  alternating  current  is  flowing  is  equal  to 


Jri2dt=r\       i2dt 
O  I/O 


A  steady  current  7  in  the  same  interval  of  time  will  develop  in 
this  same  resistance  an  amount  of  heat  energy  equal  to  rPT. 
Hence  the  alternating  current  will  develop  in  any  given  resistance 
in  any  given  time  the  same  amount  of  heat  energy  as  a  direct 
current  /  provided 


or 

I2=-\      i2dt  (6) 

The  right-hand  side  of  this  equation  represents  the  mean  of  the 
squares  of  the  instantaneous  values  of  the  alternating  current; 


ALTERNATING  CURRENTS  31 1 

hence  an  alternating  current  will  develop  the  same  amount  of 
heat  energy  in  a  given  resistance  as  a  direct  current  which  has  a 
value  equal  to  the  square  root  of  the  mean  of  the  squares  of  the 
instantaneous  values  of  the  alternating  current.  The  square  root 
of  the  mean  of  the  squares  of  the  instantaneous  values  of  an  alter- 
nating current  over  a  complete  period  is  called  the  effective  value 
of  the  alternating  current.  In  specifying  the  value  of  an  alternating 
current  as  so  many  amperes  this  effective  value  is  always  meant 
unless  specifically  stated  otherwise.  In  the  same  manner  the 
square  root  of  the  mean  of  the  squares  of  the  instantaneous  values  of 
an  alternating  potential  difference  over  a  complete  period  is  called 
the  effective  value  of  the  alternating  potential  difference.  When 
the  value  of  an  alternating  potential  difference  is  specified  as  so 
many  volts,  this  effective  value  is  always  meant  unless  specifically 
stated  otherwise. 

The  reason  for  selecting  this  particular  function  of  the  in- 
stantaneous values  of  an  alternating  current  or  potential  differ- 
ence as  the  measure  of  the  current  or  potential  difference  is  that 
the  deflection  of  all  instruments  used  in  alternating  current 
measurements  is  a  function  of  this  effective  value.  Moreover, 
Joule's  Law  for  the  heating  effect  of  a  steady  current  also  applies 
directly  to  the  heating  effect  of  an  alternating  current  provided 
the  current  is  expressed  in  terms  of  its  effective  value;  i.e.,  the 
average  power  dissipated  in  a  resistance  r,  when  an  alternating 
current  of  effective  value  /  flows  through  it,  is  rP. 

In  case  the  current  is  a  harmonic  function  of  time,  a  simple 
relation  exists  between  its  effective  and  maximum  values.  Let 
T  be  the  period  of  the  current  and  begin  counting  time  when  the 
current  is  zero  and  increasing  in  the  positive  direction.  The 
equation  of  the  current  is  then 

i=I0sin  a)t 

where  I0  is  its  maximum  value  and  o>  =  — .     The  effective  value 
of  the  current  is  then 


i  c  T 

-  I        702  sw2  a)  t  dt 


1  —  cos  2  x 
But  from  the  trigonometric  relation  that  sin7x=—          — ,  where 

x  is  any  variable,  we  have  that 


312  ELECTRICAL  ENGINEERING 

.  1      cos  2  a)  t 


2  2 

and  therefore 


T 


Hence  the  effective  value  of  the  current  is 

7=A_  (6a) 

\/2 

Similarly  the  effective  value  of  a  harmonic   electromotive  force 
which  has  the  maximum  value  E0  is 

E=^-  (66) 

V2 

and  the  effective  value  of  a  harmonic  potential  drop  which  has 
the  maximum  value  V   is 


0 


V=      ?  (6c) 


Note  that  the  effective  value  in  each  case  is  independent  of  the 
frequency  but  depends  only  upon  the  maximum  value.  The 
above  relations  hold  only  for  harmonic  functions;  when  the  current, 
electromotive  force  or  potential  drop  is  not  a  harmonic  function 
of  the  time  equations  (6)  do  not  hold.  (See  Article  176.) 

172.  The  Use  of  Alternating  Currents.  —  When  an  alternating 
electromotive  force  is  impressed  across  the  terminals  of  an  electric 
circuit  of  any  kind  an  alternating  current  of  the  same  frequency 
is  established  in  this  circuit.*  When  the  strength  of  the  current 
between  these  two  points  at  any  instant  is  i  and  the  potential 
drop  in  the  direction  of  the  current  is  v,  the  input  of  electric  energy 
between  any  two  points  1  and  2  of  the  circuit  in  any  infinitesimal 

*The  current  established  for  the  first  second  or  two  after  the  alternating 
electromotive  force  is  impressed  is  not  strictly  an  alternating  current,  since 
its  amplitude  builds  up  gradually  to  a  constant  maximum,  just  as  the  value  of 
the  current  established  in  a  circuit  by  a  continuous  electromotive  force  builds 
up  gradually  from  zero  to  a  constant  maximum,  depending  upon  the  resist- 
ance and  back  electromotive  forces  in  the  circuit.  The  current  for  the  first 
second  or  two  is  then  an  oscillating  current.  The  building  up  of  an  alternating 
current  is  discussed  in  Article  201.  For  the  present  we  shall  confine  our  atten- 
tion to  what  takes  place  in  the  circuit  after  the  current  has  become  a  true 
alternating  current,  alternating  between  constant  positive  and  negative  maxi- 
mum values. 


ALTERNATING  CURRENTS  313 

interval  of  time  dt  is  equal  to  vidt,  and  therefore  the  rate  at  which 
energy  is  transferred  to  this  part  of  the  circuit  at  this  instant,  or 
the  power  input  between  1  and  2  at  this  instant,  is  p  =vi.  Although 
the  average  values  of  an  alternating  current  and  an  alternating 
p.d.  over  a  complete  period  are  both  zero,  the  average  value  of 
the  power  input  (or  output,  when  the  potential  drop  is  in  the 
opposite  direction  to  the  current)  when  the  current  and  p.d.  have 
the  same  frequency  is  not  zero  except  in  certain  special  cases. 
Hence  an  alternating  current  may  be  used  to  transmit  electric 
energy  just  as  a  continuous  current  is  used  for  this  purpose.  In 
fact,  the  transmission  of  electric  energy  over  long  distances  can 
be  accomplished  much  more  economically  by  the  use  of  alternat- 
ing than  by  the  use  of  continuous  currents.  Alternating  current 
generators  are  much  less  expensive  for  the  same  power  output, 
and  certain  forms  of  alternating  current  motors,  particularly  the 
induction  motor,  are  cheaper  to  build  and  require  less  care  in 
operation  than  a  direct  current  motor  of  the  same  power  output. 
An  additional  advantage  of  alternating  currents  comes  from  the 
fact  that  by  means  of  a  device  called  an  alternating  current 
transformer,  power  may  be  readily  transferred  from  a  circuit  in 
which  the  current  is  small  and  the  voltage  high  to  a  circuit  in 
which  the  voltage  is  low  and  the  current  large.  Since  the  power 
lost  in  a  transmission  line  depends  upon  the  square  of  the  current, 
it  is  obvious  that  for  economical  transmission  the  current  should 
be  kept  small,  and  therefore  the  voltage  high.  On  the  other 
hand,  a  high  voltage  is  dangerous,  particularly  inside  of  buildings. 
Hence  electric  power  is  usually  generated  at  a  comparatively  low 
voltage,  "  stepped  up  ".by  means  of  a  transformer  to  a  high  volt- 
age for  transmission,  and  then  "  stepped  down  "  by  means  of 
another  transformer  to  a  comparatively  low  voltage  for  local 
distribution  and  use. 

173.  Alternating  Current  Transformer.  —  An  alternating  cur- 
rent transformer  consists  essentially  of  two  independent  windings 
wound  on  the  same  closed  iron  core.  When  a  current  is  estab- 
lished in  either  winding  a  magnetic  flux  is  established  through 
the  iron  core,  and  when  this  current  varies  with  time  the  flux 
varies  with  time  and  therefore  an  electromotive  force  is  induced 
in  each  winding.  Let  Nl  and  N2  be  the  number  of  turns  in  series 
in  the  two  windings  respectively,  and  let  (j>  be  the  number  of  lines 
of  magnetic  induction  established  in  the  core  at  any  insta-nt. 


314  ELECTRICAL  ENGINEERING 

When  all  the  lines  of  induction  link  all  the  turns  of  both  windings 
(that  is  ,  when  there  is  no  magnetic  leakage)  the  electromotive 

forces  induced  in  the  two  windings  at  this  instant  are  e1=N1  —  ? 

dt 


and  e2=N2—-  respectively.     Hence  these  electromotive  forces  are 
dt 

directly  proportional  to  the  number  of  turns  in  series  in  the  two 
windings  respectively.  When  the  terminals  of  the  first  or  primary 
winding  are  connected  to  the  terminals  of  an  alternating  current 
generator  and  the  terminals  of  the  second  or  secondary  winding 
are  connected  to  an  alternating  current  motor  or  other  receiving 
device,  energy  will  be  transmitted  through  the  transformer  to  the 
receiving  device.  Neglecting  the  dissipation  of  heat  energy  in 
the  connecting  wires  and  in  the  windings  and  core  of  the  trans- 
former, the  power  input  into  the  primary  winding  of  the  trans- 
former is  equal  to  the  power  output  of  the  secondary  winding. 
Hence  calling  t\  and  ia  the  instantaneous  values  of  the  currents 
in  the  two  windings,  wre  have  that 

BI  1i  :==C2  ^2 

e      N  i      N 

But  —  =  —  ;    therefore  —  =  —  .     That  is,  the  electromotive  forces 

e2     N2  i2     Nl  . 

induced  in  the  two  windings  are  to  each  other  as  the  number  of 
turns  in  the  respective  windings,  and  the  currents  in  the  two 
windings  are  to  each  other  inversely  as  the  number  of  turns  in 
the  respective  windings.  These  relations  are  only  approximate, 
since  the  assumed  conditions  are  only  approximately  realized  in 
practice.  For  a  full  discussion  of  the  alternating  current  trans- 
former see  any  text-book  on  alternating  current  machinery. 

174.  Average  Power  Corresponding  to  a  Harmonic  P.D.  and 
a  Harmonic  Current.  —  Power-Factor.  —  The  average  power  input 
into  a  circuit  when  a  harmonic  current  is  established  in  the  circuit 
and  a  harmonic  p.d.  is  established  between  the  terminals  of  the 
circuit  can  be  readily  expressed  in  terms  of  the  effective  values 
of  the  current  and  the  p.d.  and  the  difference  in  phase  between 
the  current  and  p.d.  Let 

i=I0sin  ((ot  +  0t) 
be  the  value  of  the  current  at  any  instant  /;  and  let 

v  =  V0sin(a)t  +  02) 
be  the  value  of  the  potential  drop  in  the  direction  of  the  current  i 


ALTERNATING  CURRENTS 


315 


at  this  same  instant.     The  power  input  into  the  circuit  at  this 
instant  is  then 

p=vi  =  V0  I0  sin  (cot  +  #,)  sin  (cot  +  02) 
From  the  trigonometric  relations  that 

cos  (a-fo)  =cos  a  cos  b—  sin  a  sin  b 
cos  (a—  b)  =cos  a  cos  b  +  sin  a  sin  b 
we  have,  subtracting  the  upper  from  the  lower,  that 

sin  a  sin  b  =  J  cos  (a—  6)  —  J  cos  (a  +  6)  (7) 

Hence,  substituting  cot  +  6^  for  a  and  cot  -f  02  for  6  we  have 

sin  (cot  +  9,)  sin  (cot  +  02)  =J  cos  (0,  -  02)  -  i  cos  (2cot  +  0,  +  02) 
Hence  the  instantaneous  power  is 


p  =vi  =        °  cos  (0,  -  ft)  -     —°  cos  (2cot  +  61 


(8) 


Therefore  the  instantaneous  power  is  also  a  harmonic  function  of 
time  but  has  twice  the  frequency  of  the  current  or  p.d.,  and  it  is 

unsymmetrical  with  respect  to  the  axis  of  time  unless  ft—  ft=— , 
i.e.,  unless  the  current  and  p.d.  are  in  quadrature.     The  curves 


Fig.  97. 

for  current,  p.d.  and  power  in  the  general  case  are  shown  in  Fig. 
97.  Note  that,  in  general,  for  part  of  the  time  the  power  is  posi- 
tive and  for  part  of  the  time  the  power  is  negative ;  whiph  means 
that  energy  is  transferred  to  the  circuit  during  part  of  each  cycle 
and  transferred  from  the  circuit  during  the  remainder  of  the  cycle. 


316  ELECTRICAL  ENGINEERING 

The   average   power   transferred   to   the   circuit   during   each 
complete  period  T  is 


!_  CT  .     =1  CTV0I0 

/T7      I  /77      I  C\ 

1 J  o  1 J  o      z 


~£)  ''  •        7  i    "__    I  /  /J  /I    \  /c\  /       I        /3          I       M  \       1    ^1 

T  J   0 

Therefore 

7  /    /  1 

P  =  -£JT  (t  cos  (9,  -92}-—  sin  (2a)t  +  0,  +  02) 


T 
o 
Whence 


and  therefore  the  average  power  input  is 

P  =  VI  608(0,-  02)  (9) 

where  V  and  I  are  the  effective  values  respectively  of  the  p.d.  and 

current,  that  is  V— — —  and  7  = — ^;  and  0!—02  is  the  difference 

V2  V2 

in  phase  between  the  current  and  p.d. 

Hence,  when  the  current  and  p.d.  differ  in  phase,  the  average 
power  input  into  the  circuit  is  less  than  the  product  of  the  effective 
current  and  the  effective  p.d.  The  ratio  of  the  true  power  input 
into  a  circuit  to  the  product  of  the  effective  current  and  effective 
p.d.  is  called  the  power  factor  of  the  circuit.  In  the  case  of  a 
harmonic  current  and  a  harmonic  p.d.  the  power  factor  is  there- 
fore equal  to  the  cosine  of  the  angle  which  expresses  the  difference 
in  phase  between  the  current  and  the  p.d.  Hence  this  angle 
is  frequently  called  the  power-factor  angle  of  the  circuit. 

Equation  (9)  may  then  be  expressed  in  words  as  "  the  average 
power  input  into  any  circuit,  when  there  is  established  a  harmonic 
current  in  the  circuit  and  a  harmonic  p.d.  of  the  same  frequency 
across  the  terminals  of  the  circuit,  is  equal  to  the  product  of  the 
effective  values  of  the  current  and  the  p.d.  times  the  power  factor 
of  the  circuit." 

When  current  and  p.d.  are  in  quadrature,  i.e.,  when  0l—02  =  ±  — 

then  cos  (Ol—  02)  =0,  that  is,  the  power  factor  is  zero,  and  the 
average  power  input  into  the  circuit  is  also  zero.  The  sine  curve 
representing  the  instantaneous  power  is  then  symmetrical  with 
respect  to  the  axis  of  time.  In  this  case  as  much  work  is  done 
on  the  current  during  one  half  of  the  cycle  of  the  power  curve  as 


ALTERNATING  CURRENTS  317 

is  done  by  the  current  during  the  other  half  of  this  cycle ;  the  total 
work  done  by  the  current  in  a  whole  cycle  is  zero.  When  current 
and  p.d.  are  in  phase,  i.e.,  when  B1  —  $2=0  then  cos  (01—02)  =1  ; 
that  is,  the  power  factor  is  unity  and  the  average  electric  power 
input  for  the  given  current  and  p.d.  is  a  maximum  and  equal  to 
VI,  where  V  and  /  are  the  effective  values  of  p.d.  and  current. 
In  this  case  the  entire  curve  of  instantaneous  power  lies  above 
the  axis  of  time.  When  current  and  p.d.  are  in  opposition,  i.e., 
when  0l—02  =  ±7T  then  cos  (01—02)=  —  l;  that  is,  the  power 
factor  is  again  numerically  equal  to  unity,  but  the  average  electric 
power  input  is  negative,  i.e.,  work  is  done  on  the  current  at  the 
average  rate  VI.  In  this  case  the  entire  curve  for  instantaneous 
power  lies  below  the  axis  of  time. 

In  general,  whenever  the  power-factor  angle  0^  —  02  is  greater 

than  +—  or  less  than  —  -,  that  is,  whenever  the  current  leads  or 
2  2 

lags  behind  the  potential  drop  in  the  direction  of  the  current  by 
more  than  90°,  work  is  done  on  the  current  at  the  average  rate 
equal  to  the  numerical  value  of  the  expression  VI  cos  (Ol—02). 
This,  part  of  the  circuit  then  acts  like  a  generator.  Let  the  in- 
stantaneous values  of  the  net  rise  of  potential  in  the  direction  of 
the  current  in  this  part  of  the  circuit,  i.e.,  the  terminal  e.  m.  /., 
be  represented  by  the  equation  e=E0  sin  (a)t  +  0f).  Then  E0  is 
numerically  equal  to  V0  and  0'  is  equal  to  02  +  ir,  for  the  rise  of 
potential  at  any  instant  is  opposite  to  the  drop  of  potential  at 
this  instant.  The  average  power  output  of  this  part  of  the  circuit 
is  then 

P0  =  EI  cos  (0,-ff)  (9a) 

where  E  =  V  is  the  effective  value  of  the  terminal  electromotive 
force.  This  output  P0  is  positive  when  (Ol—0f)  is  greater  than 

—  -  and  less  than  — .     Equation  (9)    gives   the  average   electric 
2  2 

power  input  into  the  circuit  and  equation  (9a)  gives  the  average 
electric  power  output  of  the  circuit. 

175.  Power  Corresponding  to  a  Non-Harmonic  P.D.  and  a  Non- 
Harmonic  Current.  —  When  the  p.d.  and  current  are  not  harmonic 
functions  of  time  the  above  expressions  for  power,  equations  (9), 
do  not  apply.  An  alternating  p.d.  or  current,  however,  may 
always  be  represented  by  a  Fourier's  series  (see  Article  166),  no 


318  ELECTRICAL  ENGINEERING 

matter  what  may  be  the  shape  of  the  curve  or  "  wave  "  represent- 
ing it.     For  example,  let  the  current  and  p.d.  be 

i  =/!  sin  (a) *  +  00  +  Iz  sin  (2cot  +  02)  +  73  sin  (3a)t  +  0a) 
v  =  V,  sin  (o)t  +  0'0  +  V2  sin  (2<ot  +  0'2)  +  V3  sin  (3o)t  +  0'3) 
In  this  case  the  instantaneous  power  is  (see  equation  7) 

w=i  [VJt  cos  (0,-  0'0  +  V2I2  cos  (02-  0'2)  +  V3I3  cos  (03-  0'3) 
-  VJ,  cos  (2o>*+  ft  +  0'0  -  V2I2  cos  (4  o>£+  02+  0'2) 

-  V3I3  cos  (6orf+  03+  0'3)  +  VJ,  cos  (o>t+0'2  -  0,) 
+  V3I,  cos  (2o>t  +  0'3-  9,)  +  VJ2  cos  (a)t+02-  0'0 
+  V3I2  cos  (cot  +  03f -  02)  +  VJ3  cos  (2<ot  +03-  0\) 
+  72/3  cos  (cot  +  03-  0'2)  -  V2I,  cos  (3orf  +  0,  +  0'2) 

-  VJ,  cos  (4o>t  +  0,  +  0'3)  -  VJ2  cos 

-  V3I2  cos  (5o)t+02+0'3)  -  VJ3  cos 

-  V2I3  cos(5<ot+03+0'2)] 

The  average  of  this  instantaneous  power  for  a  complete  cycle  of 

the  fundamental  period  T— —  is 

CD 


1  r T 

=  T         mdf' 
J  o 


P=- 
Hence  the  average  power  is 

P=Y^lCos(Ol-9'l)+^  cos  (02-02')  +^  cos  (03 -0,')   (10) 
2i  2i  Zi 

since  the  integral  between  the  limits  0  and  T  for  each  of  the  har- 
monic terms  containing  t  in  the  equation  for  instantaneous  power 
is  zero.  Hence  when  there  exists  in  a  circuit  an  alternating 
current  and  an  alternating  p.d.  of  any  kind  whatever,  the  average 
power  is  equal  to  the  sum  of  the  values  of  the  average  power 
corresponding  to  each  pair  of  harmonics  of  the  same  frequency 
which  exist  in  the  Fourier's  series  for  the  current  and  p.d.  That 
is,  the  average  power  input  corresponding  to  each  pair  of  har- 
monics of  the  same  frequency  is  independent  of  what  other  har- 
monics may  be  present.  Note  that  when  any  harmonic  is  absent 
in  either  the  p.d.  or  the  current  this  harmonic  contributes  nothing 

to  the  average  power.     For  example  the  term       — -  cos  (B3—6'^ 

equals  zero  for  either  73=0  or  V3=0. 

It  should  also  be  remembered  that  in  most  practical  cases  the 
curves  representing  p.d.  and  current  are  each  symmetrical  with 


ALTERNATING  CURRENTS  319 

respect  to  the  axis  of  time,  and  therefore  as  a  rule  only  the  odd 
harmonics  are  present. 

176.   Effective  Value  of  a  Non-Harmonic  Current  or  P.D.  — 

When  the  current  is  of  the  form 

i  =7j  sin  (a)t  +  0j)  +  72  sin  (2  a)t  +  02)  +  73  (sin  3  cot  +  03} 

1  C  T 
the  square  of  the  effective  value  is,  by  definition,  72=—  I      i2dt. 

That  is,  the  square  of  the  effective  value  of  the  current  is  of 
the  same  mathematical  form  as  the  average  power  correspond- 
ing to  the  current  i  and  an  equal  p.d.  in  phase  with  it.  Hence 
from  equation  (10) 

r  2  T  2  72 

J2  _*1_  -j_  ll_  +  _  -I 

cy  c\  o 

But  — l—,  — 2-,  — L,  etc.,  are  the  effective  values  of  the  harmonics 

V2   V2  V2 

which  have  the  maximum  values  I1}  72,  73,  etc.,  respectively. 
Hence,  the  effective  value  of  any  non-harmonic  current  is  equal  to 
the  square  root  of  the  sum  of  the  squares  of  the  effective  values  of 
all  the  harmonics  present  in  the  current  wave.  Similarly,  the 
effective  value  of  any  non-harmonic  p.d.  is  equal  to  the  square 
root  of  the  sum  of  the  squares  of  the  effective  values  of  all  the 
harmonics  present  in  the  p.d.  wave.  When  Ilf  72,  73,  etc.,  are 
taken  to  represent  the  effective  values  of  the  various  harmonics 
of  the  current  and  V1}  F2,  F3,  etc.,  are  taken  to  represent  the 
effective  values  of  the  various  harmonics  of  the  p.d.}  the  effective 
value  of  the  resultant  current  may  be  written 

and  the  effective  value  of  the  p.d.  may  be  written 

~~f  F,2+F,2  +  -=^~  (Ha) 


and  the  average  power  may  be  written 

P  =I1V1  cos  0t  +  7272  cos  02  + 7373  cos  0,+  -  (116) 

where  0l}  02,  03,  etc.,  are  respectively  the  differences  in  phase 
between  the  harmonics  of  current  and  p.d.  of  the  same  frequency. 
177.  Equivalent  Sine- Wave  P.D.  and  Current.  —  In  practical 
work  the  p.d.  and  current  are  seldom  simple  harmonic  functions 
of  true,  i.e.,  are  seldom  "  sine  waves/'  but  each  contains  one  or 
more  of  the  odd  harmonics.  As  a  rule,  however,  it  is  not  neces- 
sary to  consider  these  harmonics  separately,  but  as  a  first  ap- 
proximation the  p.d.  and  current  may  each  be  considered  as  a 


320  ELECTRICAL  ENGINEERING 

sine  wave  having  an  effective  value  equal  to  the  effective  valuo 
of  the  actual  wave  and  differing  in  phase  by  an  angle  6  such  that 
VI  cos  6  represents  the  average  power,  where  V  and  /  are  the 
effective  values  of  the  true  p.d.  and  current  waves  respectively. 
In  certain  special  cases,  however,  it  is  necessary  to  analyse  the 
waves  into  their  constituent  harmonics. 

178.  Determination  of  the  Maximum  Value  and  Phase  of  the 
Harmonics  in  a  Wave  of  Any  Shape.  —  When  the  wave  shape  of 
a  p.d.  or  current  is  known,  the  effective  value  and  the  phase  of 
the  harmonic  of  any  order  may  be  readily  determined.  For 
example,  the  instantaneous  value  of  the  current  may  be  written 

i  =1,  sin  x  +  I2  sin  (2x+  02)  +  13  sin  (3x  +  03)  +  etc.  (a) 
where  x=27r  ft  and  /  is  the  frequency  of  the  wave.  The  in- 
stantaneous value  of  the  current  i  corresponding  to  any  value  of 
x  is  also  given  by  the  corresponding  ordinate  of  the  curve  repre- 
senting the  current  wave.  We  wish  to  determine  the  constants 
rv  72,  73,  etc.,  and  02,  03,  etc.,  which  will  make  the  curve  rep- 
resented by  (a)  coincide  with  the  actual  current  wave.  Con- 
sider, for  example,  the  third  harmonic.  Multiply  equation  (a) 
by  sin  3x  and  integrate  with  respect  to  x  over  an  entire  period  of 
the  wave,  i.e.,  between  the  limits  x  =o  and  x  =2  IT.  We  then  have 

J27T  f*2ir 

i  sin  3x  dx  =    I       [7j  sin  x  sin  3x  +  72  sin  (2x  +  02)  sin  3x 
o  J  o 


+  I3sin  (3x  +  0.3)  sin3x  +  etc.]  dx 

But  the  integral  of  each  term  in  the  right-hand  member  of  this 
equation  between  the  limits  o  and  2  77  is  zero,  except  for  the 
particular  term  7  sin  (3x  +  0)  sin  3x,  the  integral  of  which  from 

0  to  2  TT  is  -  ^  cos  #3  =7r73  cos  03.     (See  Article  175.)     Hence 


2ir 

i  sin  3x  dx  =  TT  73  cos  03 
o 


i  sin  3x  dx  may  be  determined  graphically 

9 

by  plotting  the  expression  i  sin  3x  or  ordinates  against  x  as 
abscissas,  and  determining  the  area  of  this  curve  by  means  of  a 
planimeter.  Call  this  area  A3,  then 

A3  =  7rI3cos03  (6) 


ALTERNATING  CURRENTS 


321 


Next,  multiply  the   equation  (a)    by  cos3x=sin 

exactly  the  same  manner  we  then  have  that 

r  I*        \ 

i  cos  3x  cte  =7T/3  cos      -03    =TT  I3  sin  0 


in 


£1* 
f  I        i 
J  o 


and  the  value  of  I        i  cos  3x  dx  may  be  determined  graphically 


as  in  the  first  case  by  plotting  the  curve  i  cos  3x  and  finding  its 
area  by  means  of  a  planimeter.     Call  this  area  B3}  then 

B3  =  7rI3sin  03  (c) 

From  equations  (6)  and  (c)  we  then  have 

2+B*  (12a) 


7T 


and 


*.-*,»' 


(126) 


Note  that  in  case  the  wave  is  symmetrical  with  respect  to  the 
axis  of  time  it  is  unnecessary  to  look  for  the  even  harmonics. 
Also,  when  the  wave  is  symmetrical,  the  curves  i  sin  3x  and 
i  cos  3x  need  be  plotted  for  only  a  half  period  of  the  wave.  Call- 
ing 0,3  and  ft3  the  areas  of  these  curves  for  half  a  period  of  the 
wave,  we  then  have 


(136) 


This  method  of  determining  •  the  harmonics  present  in  the 
wave  is  of  course  applicable  to  the  determination  of  the  harmonic 
of  any  order. 


Fig.  98. 


Example:  Suppose  the  current  is  a  symmetrical  rectangular 
wave  as  shown  in  Fig.  98.     Let  /  be  the  value  of  the  current 


322  ELECTRICAL  ENGINEERING 

during  the  positive  half  of  the  wave.  Since  the  wave  is  sym- 
metrical, only  the  odd  harmonics  can  be  present.  Consider  the 
nth  harmonic,  where  n  is  any  odd  positive  integer.  Then 

r  f     I  Y  21 

an=  \      I  sin  nx  dx  —  I  —  —  cos  nx  }  = — • 
V     n  /„    n 

J  o 

and  f*  f1  Y 

/3n  =  I       /  cos  nx  dx  =  I  -  sin  nx]    =0 

Jo  o 

since  n  is  odd.     Hence  from  equations  (13) 


and 


n 

Hence  all  the  odd  harmonics  exist  in  a  symmetrical  rectangular 
wave,  the  maximum  values  of  the  harmonics  varying  inversely 
as  their  order  n.  A  symmetrical  rectangular  wave  having  a  maxi- 
mum value  /  may  then  be  represented  by  the  infinite  series 


_ 
i— 


n 

n=l 

where  n  has  all  the  odd  values  from  1  to  oo  and  /  is  the  frequency 
of  the  wave. 

179.  The  Fisher-Hinnen  Method  for  Analysing  a  Non-Harmonic 
Wave.  —(See  Electric  Journal,  Vol.  5,  p.  386  and  Elektrotech- 
nische  Zeitschrift,  May  9,  1901.)  This  method  is  much  simpler 
than  that  described  above,  exc.ept  in  the  rare  cases  where  the 
resultant  wave  may  be  represented  by  a  simple  integrable  func- 
tion. The  method  is  based  on  the  following  facts  : 

1.  The  algebraic  sum  of  any  n  equally  spaced  ordinates  of  a 

k 
sine  wave,  when  these  ordinates  are  spaced  —  th  of  a  wave  length 

Tb 

apart,  where  k  is  any  integer  which  is  not  a  multiple  of  n,  is  zero. 

2.  The  algebraic  sum  of  n  ordinates  of  a  sine  wave  when  these 

k 

ordinates  are  spaced  —  wave  lengths  apart,  where  k  is  a  multiple 
n 

of  n,  is  equal  to  n  times  the  ordinate  of  this  wave  at  any  one  of 
these  points. 

3.  The  maximum  ordinate  of  any  sine  wave  is  equal  to  the 
square  root  of  the  sum  of  the  squares  of  any  two  ordinates  spaced 
a  quarter  of  a  wave  length  apart. 


ALTERNATING  CURRENTS  323 

4.  Let  ?/!  be  the  ordinate  of  a  sine  wave  at  any  point  at  an 
angular  distance  xl  from  the  origin,  and  let  y2  be  another  ordinate 
of  this  wave  a  quarter  of  a  wave  length  to  the  right  of  xt.  Then 
the  angular  distance  measured  from  the  point  xl  toward  the  left 
to  the  point  at  which  this  wave  first  crosses  the  x  axis  in  the 
positive  direction  is 


Consider  a  wave  of  any  form  whatever  (for  example,  the 
wave  shown  in  Fig.  95)  and  let  the  highest  harmonic  of  this 
wave  be  the  nth.  Let  yly  yz,  y5,  -------  y  2n-i  be  n  ordinates  of 

360 
this  wave  spaced  —    degrees  apart,  where  360°  corresponds  to 

a  complete  wave  length  of  the  given  wave.  Let  a/,  a3',  as',  be 
the  corresponding  ordinates  of  the  fundamental  or  first  harmonic 
of  this  wave,  a/',  a"  ,  a/,  be  the  corresponding  ordinates  of  the 
second  harmonic  of  this  wave,  a"'  ',  a/',  a/",  etc.,  be  the  cor- 
responding ordinates  of  the  third  harmonic,  and  so  on;  the  cor- 
responding ordinates  of  the  nth  harmonic  being  a^n\  a3(n), 
a5(n),  etc.  We  then  have  that 

=  '         "     ------------     n) 


=  a2n_/  +  a,n_/'  f  a2n_/"+ 


The  ordinates  a/  to  a2n_/  of  the  first  harmonic  are  ordinates 
of  a  sine  wave  and  are  spaced  —  th  of  a  wave  length  apart,  and 

therefore,  from  Proposition  1,  their  sum  is  zero.  Similarly,  the 
ordinates  «/'  to  «2n-i"  are  ordinates  of  a  sine  wave  of  half  the 
wave  length  of  the  fundamental  and  therefore  the  angular  dis- 

2 
tance  between  these  ordinates  is  —  th  of  the  wave  length  of  the 

n 

sine  wave  of  which  they  are  the  ordinates  ;  hence  the  sum  of  the 
ordinates  a/'  to  a2n-/'  is  zero.  Similarly  for  all  the  other  ordinates 

except  those  of  the  nth  harmonic.     The  latter  are  spaced  —  =  1 

wave  length  apart,  and  therefore,  from  Proposition  2,  their  sum 
is  equal  to  n  times  the  value  of  the  ordinate  of  this  harmonic 
at  any  one  of  the  points  1,  3,  5,  etc.  Hence  the  value  of  the 
ordinate  of  the  nth  harmonic  at  the  point  1  is 

^i  =-  (2/1  +  2/3  +  2/5  +  ------------  2/2n.i) 

n 

Similarly,  if  y2,  y4,  y6,  ---------  y2H  are  ordinates  of  the  given 

wave  one  quarter  of  a  wave  length  of  the  nth  harmonic  to  the 


324  ELECTRICAL  ENGINEERING 

right  of  the  first  set  of  ordinates,  their  sum  will  be  equal  to  n 
times  the  ordinate  of  the  nth  harmonic  a  quarter  of  a  wave  length 
of  this  harmonic  from  yl ;  call  this  ordinate  Bn,  then 

Bn  =-  (2/2  +  2/4  +  2/6  + y2n) 

n 

Then  from  Proposition  3,  the  maximum  value  of  the  ordinate 
of  the  nth  harmonic  is 

Y  —\/A  2  +  B  2 

1  n  —  v  ^n    '  °n 

From  Proposition  4,  the  angular  distance  to  the  left  of  yt  at  which 
this  harmonic  cuts  the  x  axis  in  the  positive  direction  is 

f*-**4^5 

Bn 

when  360  degrees  are  taken  equivalent  to  a  wave  length  of  the 
nth  harmonic.  When  360  degrees  are  taken  equivalent  to  a  wave 
length  of  the  given  wave  this  angular  distance  is 

<^n=Iton-1— n 
n         Bn 

Consider  next  the  mth  harmonic,  and  erect  two  sets  of  m 
ordinates,  the  ordinates  of  each  set  being  spaced  360°  apart 
(considering  a  wave  length  of  the  given  wave  as  equivalent  to 
360°)  and  the  second  set  a  quarter  of  a  wave  length  of  this  har- 
monic to  the  right  of  the  first  set.  Then,  if  the  harmonics  of  higher 
order  are  not  multiples  of  m,  we  have  as  before  that  the  ordinate 
of  the  mth  harmonic  at  the  point  1  is 

Am  =—  (2A  +  2/3  +  2/5 2/2m-0 

m 

and  the  ordinate  of  the  mth  harmonic  at  the  point  2,  which  is  a 
quarter  wave  length  of  this  harmonic  to  the  right  of  1 ,  is 

Bm  =—  (2/2+  2/4  +  2/5  + 2/2m) 

m 

Whence  the  maximum  value  of  this  harmonic  is 


Y    —\/A   24-B  2 
1  m  —  v  **m  T  £>m 

and  it  cuts  the  x  axis  at  the  angular  distance 


m          Bm 

to  the  left  of  the  first  ordinate  when  360  degrees  are  taken  equiv- 
alent to  the  wave  length  of  the  original  wave. 

If  there  also  exists  in  the  given  wave  a  harmonic  of  the  nth 
order,  where  n  is  a  multiple  of  m,  that  is  if  n=km,  where  k  is  an 
integer,  then  from  Proposition  2,  since  each  set  of  these  m  ordinates 

97 

is  spaced  —  =k  wave  lengths  of  the  nth  harmonic  apart,  we  have 
that  the  sum  of  the  first  set  of  m  ordinates  also  contains  m  times 


ALTERNATING  CURRENTS  325 

the  ordinate  of  the  nth  harmonic  at  the  point  1.      Calling  An 
the  ordinate  of  the  nth  harmonic  at  the  point  1  we  then  have 
that 


=-  (yl  +  2/3+  2/5  + +  2/m-O  -  An' 

m 


Similarly, 
B 


m=- 

m 


where  Bn'  is  the  ordinate  of  the  nth  harmonic  at  the  point  2. 

A  similar  correction  must  be  applied  for  all  other  harmonics 
of  higher  order  than  the  rath  if  the  orders  of  these    harmonics 
are  multiples  of  m. 

The  waves   of  current  and  electromotive  force  with  which 
one  has  to  deal  in  practice  usually  contain  only  the  odd  har- 
monics; also,   as   a  rule,  the  harmonics   of   higher  orders  than 
the  seventh  are  negligible.     In  this  case  the  three  harmonics, 
the  third,   fifth,  and  seventh,  are  not  multiples   of  each  other 
and  consequently  no  correction  term  has  to  be  applied.     More- 
over, it  is  sufficient  to  consider  the  ordinates  of  only  half  a  wave 
length.     To  determine  the  third  harmonic  divide  the  base  line 
of  this  half  wave  into  2n=6  equal  parts  and  measure  the  ordinates 
at  the  beginning  of  each  of  these  six  segments.     Let  these  ordi- 
nates be  2/t,  2/2,  ......  -  -2/6.    Let  the  beginning  of  the  first  segment 

be  taken  where  the  given  wave  cuts  the  x  axis  and  call  this  point 
the  origin,  then  yl  =0  and  we  have 


B3=-  (2/2+2/6-2/4) 

o 

Then  the  maximum  value  of  the  third  harmonic  is 


and  it  cuts  the  x  axis  at  the  angular  distance 


to  the  left  of  the  origin.     The  equation  of  the  third  harmonic 
is  then 

y3=Y3  sin  3  (x+<f>3) 

Similarly,  starting  at  the  same  point  and  dividing  the  half  wave 
into  2n  -10  segments,  we  have  for  the  5th  harmonic 

A5  =  -  (2/5+2/0-2/3-2/7) 

o 

B6=-  (2/2  +  2/6  +  2/10-2A-2/8) 
5 


326  ELECTRICAL  ENGINEERING 


*-=Iw'(l) 

and  the  equation  of  this  harmonic  is 


The  determination  of  the  seventh  harmonic  is  carried  out  by  an 
exactly  similar  process,  dividing  the  base  line  into  2n=l4  equal 
parts. 

The  ordinates  of  the  fundamental  at  the  origin  and  at  the 
point  corresponding  to  one  quarter  of  a  wave  length  from  the 
origin  are  then  respectively 

•"-i  ~       -A  3      "-a      A-t 

and  B^yn+Bt-Bt  +  B? 

where  ym  is  the  ordinateof  the  given  wave  corresponding  to  the 
point  a  quarter  of  a  wave  length  from  the  origin  (i.e.,  the  mid- 
ordinate  of  the  given  wave)  and  A3,  A5,  A7,  B3,  B5,  and  B7  are 
the  quantities  above  determined.  The  maximum  value  of  the 
fundamental  is  then 


and  it  cuts  the  axis  of  x  at  the  angular  distance 


to  the  left  of  the  origin.     Its  equation  is  therefore, 

yl  =  Yl  sin(x+fa) 

The  equation  of  the  given  wave  is  then 
y  =  Yl  sin  (x+  </>!)  +  Y3  sin  3  (x+fa)  +  Y5  sin  5  (x+  (f>5) 

+  Y7sin7  (x+fa)  (I4a) 

The  effective  value  of  the  given  wave  is 

*;  2  (146) 

and  the  average  value  is 

^aver=-    Y^os^^ -Y3  cos  3  fa  +  - Y5  cos5fa+-Y7cos7fa 
77"L  o  5  7  J 

a*o 

In  employing  the  above  formulas  strict  attention  must  be  paid 
to  algebraic  signs. 

Example.  In  the  curve  shown  in  the  figure  the  ordinates  at 
6  equally  spaced  points,  starting  from  the  point  where  the  curve 
cuts  the  base  line  are 

yl  =0  y4  =940 


7/3=660 


ALTERNATING  CURRENTS 


327 


and  the  ordinates  corresponding  to  10  equally  spaced  points, 
starting  from  the  point  where  the  curve  cuts  the  base  line  are 
2/1=0  2/3=719  2/5=702  y7=W8Q  y,  =639 

2/2=470  2/4=678  2/6=940  ys=92Q  2/10=375 


Fig.  99. 

From  the  first  set  of  ordinates  we  have  for  the  third  harmonic 


Y3 


>3  =  1  tan' 


96.7 


(96.7)2=150 
7        '-16.6° 


From  the  second  set  of  ordinates  we  have  for  the  fifth  harmonic 
702  +  630-719-1086 


=  -92.; 


470  +  940  +  375-678-920 


5  =V(92.8)2  +  (37.4)2  =100 


For  the  fundamental  we  then  have 

A1=114.7  +  92.8  =  -21.9 
B^  =940  +  96.7-  37.4  =  +  999.3 

Yl  =\/(21.9)2  +  (999.3)2  =999.5 

—  91  Q 

4>l=tan-1  =-1.25° 

999.3 


328  ELECTRICAL  ENGINEERING 

The  equation  of  the  given  wave  is  then 

y  =999.5  sin  (x-  1.25°)  +  150  sin  3  (x  +  16.6°) 

+  100  sin  5  (x-  13.6°) 
Its  effective  value  is 

F 
and  its  average  value 

3 

180.  Form  Factor.  —  The  form  factor  of   a  wave  is   defined 
as  the  ratio  of  the  effective  value  to   the  average  value.     For 

a   sine   wave    the   form  factor   is—-  %--  —  —  =—-—=  =  1.11.       The 


form   factor   of   a   flat-topped  wave  is  less;   of  a  peaked  wave 

718 
greater.     The  form  factor  of  the  wave  shown  in  Fig.  99  is  - 

1.068. 

181.  Amplitude  Factor.  —  The  amplitude  factor  of  a  wave  is 
defined  as  the  ratio  of  the  maximum  value  to  the  effective  value. 
For  a  sine  wave  the  amplitude  factor  is  1.414.     The  amplitude 
factor  of  the  wave  shown  in  Fig.  99  is  1.53. 

182.  Power  and  Reactive  Components  of  P.D.  and  Current— 
Whenever  we  have  to  deal  with  two  or  more  harmonic  functions 
of  time,  we  may  as  a  matter  of  convenience  begin  counting  time 
at  an  instant  when  one  of  these  functions  is  zero  and  is  increasing 
in  the  positive   direction.     For   example,   when   the   current   is 
i=I0  sin  (<0l-f-0i)  and  the  p.d.  v  =  V0  sin  (<u£  +  02),  we  may  begin 
counting  time  at  the  instant  when  v=0  and  is  increasing  in  the 
positive  direction  ;  02  is  then  equal  to  zero.     Hence,  dropping  the 
subscript  from  0V  and   also  writing  70=\/2  I  and   V0=\/2    V, 
where  I  and  V  are  the  effective  values  of  i  and  v  respectively, 
we  have 

v=V2  V  sin  ait 
i=V2  I  sin  (ut  +  0) 
The  average  power  is  then 

P  =  VIcos0  (15) 

From  the  trigonometric  formula  that 

sin  (a  +  b)  =sin  a  cos  b  +  cos  a  sin  a 
we  may  write  the  current 

i  =\/2  /  cos  0  sin  a)  t  +  \/2  I  sin  0  cos  cot 
that  is,  we  may  consider  i  as  made  up,  of  two  components 


ALTERNATING  CURRENTS  329 

i*!  =  V2  7  cos  9  sin  cot 
i2  =V/2  7  sin  9  cos  cot  =\/2  7  sw  9  sin 


The  first  component  ^  is  in  phase  with  the  p.d.  and  has  the  effective 
value  7  cos  0  ;  the  second  component  is  in  quadrature  ahead  of 
the  p.d.  and  has  the  effective  value  7  sin  0.  The  average  power 
corresponding  to  the  first  component  i,  is 

V  (7  cos  9)  cos  0  =  77  cos9  =  P 
and  the  average  power  corresponding  to  the  second  component  ?'2  is 


7T 


7  (I sin  9)  cos-=0 
45 

Hence  the  average  power  of  a  harmonic  p.d.  and  current  is  equal 
to  the  effective  value  of  the  p.d.  times  the  effeetive  value  of  the  com- 
ponent of  the  current  in  phase  with  the  p.d.  The  component  of  the 
current  in  phase  with  the  p.d.  is  therefore  called  the  power  com- 
ponent of  the  current.  The  component  of  the  current  in  quadra- 
ture with  the  p.d.  is  called  the  reactive  component  of  the  current, 
since  the  power  input  corresponding  to  this  component  of  the 
current  during  each  quarter  cycle  is  exactly  equal  to  the  power 
given  back  during  the  following  quarter  cycle. 

Similarly,  we  may  consider  the  p.d.  as  made  up  of  two  com- 
ponents 

Vl  =\/2V  cos  9  sin  (cot  +  0) 
V2  =\/2V  sin  9  cos  (cot  +  0)  =V2V  sin  9  sin  [(cot  +  9)  -  -1 

The  first  component  vl  is  in  phase  with  the  current  and  has  the 
effective  value  V  cos  9 ;  the  second  component  v2  is  in  quadrature 
behind  the  current  and  has  the  effective  value  V  sin  9.  The 
average  power  corresponding  to  the  first  component  vl  is 

7  (V  cos  9)  cos  0=7  V  cos9  =  P 

and  the  average  power  corresponding  to  the  second  component 
v2  is 

I  (V  sin  9)  cos-=0 

Hence  the  average  power  of  a  harmonic  p.d.  and  current  is  also 
equal  to  the  product  of  the  effective  value  of  the  current  times  the 
effective  value  of  the  component  of  the  p.d.  in  phase  with  the  current. 
The  effective  value  of  the  component  of  the  p.d.  in  phase  with 


330  ELECTRICAL  ENGINEERING 

the  current  is  therefore  called  the  power  component  of  the  p.d.; 
and  the  effective  value  of  the  component  of  the  p.d.  in  quadrature 
with  the  current  is  called  the  reactive  component  of  the  p.d.  The 
reactive  component  of  the  p.d.  is  in  quadrature  behind  the  current 
when  the  resultant  current  leads  the  resultant  p.d.  and  is  in 
quadrature  ahead  of  the  current  when  the  resultant  current  lags 
behind  the  resultant  p.d. 

The  name  "  wattless  component "  is  sometimes  used  for 
reactive  component,  since  the  average  watts  corresponding  to 
this  component  of  current  or  p.d.  is  zero ;  the  instantaneous  watts 
corresponding  to  this  component,  however,  are  not  zero.  Hence 
the  adjective  "  wattless  "  is  misleading. 

183.  Apparent  Power.  —  The  product  of  the  effective   value 
V  of  the  resultant  p.d.  and  the  effective  value  I  of  the  resultant 
current  is  called  the  apparent  power,  that  is 

Apparent  Power  =VI  (16) 

From  equation  (15),  we  have 

Average  Power  =  VI  cos  0 

where  0  is  the  difference  in  phase  between  p.d.  and  current. 
Hence  « 

Average  Power  /, 

Power  Factor  =—  -  =cos  V 

Apparent  Power 

The  terms  volt-amperes  and  apparent  watts  are  also  frequently  used 
for  apparent  power;  i.e.,  volt-amperes  or  apparent  watts  =  (effec- 
tive p.d.  in  volts)  X  (effective  current  in  amperes). 

184.  Reactive    Power.  —  The    expression   reactive    power    is 
used  for  the  product  of  the  effective  current  and  the  effective 
value  of  the  p.d.  in  quadrature  with  it ;  or,  what  amounts  to  the 
same  thing,  the  product  of  the  effective  p.d.  and  the  effective 
value  of  the  component  of  the  current  in  quadrature  with  the  p.d. 
That  is 

Reactive  Power  =  VI  sinO  (17) 

where  V  and  /  are  the  effective  values  of  p.d.  and  current  respec- 
tively, and  6  is  the  power-factor  angle.  From  equations  (15), 
(16)  and  (17)  it  follows  that  the  apparent  power  is  equal  to  the 
square  root  of  the  sum  of  the  squares  of  the  average  power  and 
the  reactive  power,  i.e., 

VI  =V(77  cos  0)2+  (VI  sin  0)2 


ALTERNATING  CURRENTS  331 

186.  Addition  of  Alternating  Currents  and  of  Alternating 
Potential  Differences.  —  In  any  technical  problem  that  has  to 
do  with  the  generation,  distribution  or  utilisation  of  electric 
energy  by  means  of  alternating  currents  it  is  of  fundamental  im- 
portance to  be  able  to  predetermine  the  distribution  of  the  cur- 
rents and  potential  drops  in  the  various  windings  of  the  machines 
and  in  the  transmission  line  or  network  which  forms  the  distribu- 
tion system.  The  fundamental  principles  involved  in  such  calcu- 
lations are  the  same  as  in  the  case  of  continuous  currents,  that  is, 
the  two  principles  known  as  Kirchhoff's  Laws.  However,  in  the 
ease  of  alternating  currents,  Kirchhoff's  Laws  in  the  simple  form 
in  which  they  are  stated  in  Article  98  apply  only  to  the  instan- 
taneous values  of  the  currents  and  electromotive  forces;  they  do  not 
apply  to  the  effective  values  of  these  quantities.  Effective  values 
of  alternating  currents  and  electromotive  forces  are  not  additive. 
Effective  values  of  alternating  currents  and  electromotive  forces 
are  like  the  numerical  values  of  continuous  currents  and  electro- 
motive forces ;  we  cannot  say,  when  two  batteries  are  connected 
in  a  circuit,  whether  the  net  electromotive  force  in  the  circuit 
will  be  the  sum  or  the  difference  of  the  electromotive  forces  of 
the  two  batteries  unless  we  know  their  direction  with  respect  to 
each  other.  Similarly,  when  two  alternating  electromotive  forces 
are  acting  in  the  same  circuit,  we  cannot  say  what  will  be  the  net 
electromotive  force  in  the  circuit  unless  we  know  their  directions 
with  respect  to  each  other.  The  numerical  value  of  an  alternat- 
ing electromotive  force  is  specified  by  its  effective  value,  and  the 
difference  in  direction  between  two  alternating  electromotive 
forces  is  specified  by  their  difference  in  phase.  Hence,  to  deter- 
mine the  effective  value  of  two  alternating  electromotive  forces,  it  is 
necessary  to  know  both  ihdr  effective  values  and  their  difference  in 
phase.  Similarly,  to  determine  the  effective  value  *o/  the  total 
current  leaving  any  junction  in  a  network  of  circuits,  it  is  necessary 
to  know  not  only  the  effective  values  of  the  currents  coming  up  to 
that  junction  but  to  know  also  the  phase  relations  of  these  currents. 

Consider  first  two  harmonic  alternating  electromotive  forces  of 
the  same  frequency, 

el  =  El  sin  a>  t 
and 

ez  =  E2  sin  (a)  t  +  0 ) 
in  series  between  any  two  points  of  a  circuit.     These  two  electro- 


332  ELECTRICAL  ENGINEERING 

motive  forces  then  differ  in  phase  by  0°.  If  0  =  0,  the  two  act 
in  the  same  direction  at  each  instant;  if  0  =  180°  the  two  oppose 
each  other  at  each  instant;  if  0  has  any  other  value  the  two 
electromotive  forces  act  together  during  part  of  each  cycle  and 
oppose  each  other  during  the  rest  of  the  cycle.  In  any  case,  the 
net  or  resultant  electromotive  force  at  any  instant  is 

e  =el -f- e2  =  El  sin  a)t-\-  E2  sin  (a)t-\-  0) 

The  expression  #,  sin  a)t  +  E2  sin  (a)t  +  0)  may  be  put  equal  to 
E0  sin  (a)t  +  00]  where  the  constants  E0  and  00  may  be  deter- 
mined from  the  condition  that  the  relation 

Et  sin  a) t  +  E2  sin  (a) t -  f- 0 )  =  E0  sin  (a)t-\-00) 
must  hold  at  all  times.     For  t  =  Q,  we  then  have  that 

E2sin0=E0sin00  (a) 

and  for  /  = — , 
2a) 

E!  +  E2  COS  0  =  E0  COS  00  (b) 

since  sin    ( -  +  0]  =  cos  0  and  sin  I  -  +  00 }  =cos  00 

Squaring  the  equations  (a)  and  (b)  and  adding,  we  get 
E2  +  E2  cos2  0  +  2E,E2  cos  0  +  E2  sin2  0  =  E2  cos2  00  +  E2  sin2  00 

whence 

E02=E2+E2  +  2ElE2cos  0  (18a) 

Dividing  the  equation  (a)  by  the  equation  (b),  we  get 

/)          E2  sin  0  ,..  „,  N 

tan  00= (186) 

Therefore 

Si  +  e2  =  E0  sin  (a)  t  +  00) 

where  E0  is  the  maximum  value  of  the  resultant  e.m.f.  and  is 
given  by  equation  (18a)  and  00  is  the  angle  by  which  the  re- 
sultant e.  m.  /.  leads  ev  and  is  given  by  equation  (186). 

Exactly  similar  relations  hold  for  two  harmonic  currents  of 
the  same  frequency,  and  in  fact  for  any  two  harmonic  functions 
of  the  same  frequency  in  the  same  independent  variable. 

186.  Representation  of  a  Harmonic  Function  by  a  Rotating 
Vector.  —The  results  just  deduced  may  be  arrived  at  graphically 
by  a  very  simple  method.  Let  OQ  in  Fig..  100  be  any  line  fixed 
in  the  plane  of  the  paper  and  let  the  line  OP,  be  equal  in  length 
to  Elt  and  be  pivoted  at  0  and  rotate  about  0  with  an  angular 


ALTERNATING  CURRENTS 


333 


velocity  <w ;  let  this  rotating  line  coincide  with  the  line  of  reference 
OQ  at  time  2  =  0.  Then  at  any  instant  of  time  t,  this  line  will 
make  an  angle  cot  with  OQ  and  the  instantaneous  value  of  the 
e.  m.  (.  6i  =  E!  sin  cot  at  any  instant  will  be  equal  to  the  vertical 
distance  from  Pl  to  OQ.  Similarly,  if  OP2  is  a  second  line 
equal  in  length  to  E2  rotating  about  0  with  the  same  angular 
velocity  a),  and  making  the  angle  9  with  OQ  at  time  Z  =  0,  then 
the  instantaneous  value  of  the  e.  m.  f.  e2  =  E2  sin  (a>t  +  0)  at  any 
instant  will  be  equal  to  the  vertical  distance  from  P2  to  OQ. 
Since  both  lines  rotate  with  the  same  velocity  the  angle  between 
the  two  will  be  equal  to  9  at  all  times ;  that  is  the  phase  angle 
9  measures  the  difference  in  direction  between  the  two  rotating 
vectors  representing  the  two  e.  m.  f.'s. 


O 


I       Q, 


Fig.  100. 

Let  OP0=E0  be  the  vector  sum  of  OP,  and  OP2.  It  is  then 
evident  from  the  diagram  that  the  instantaneous  value  of  (ev-\-e^ 
at  any  instant  is  the  vertical  distance  from  P0  to  OQ;  also  that 
this  line  OP0  remains  fixed  in  length  and  fixed  in  position  with 
respect  to  0  P,  and  0  P^  and  is  equal  numerically  to  the  maximum 
value  of  e!  +  e2,  and  at  any  instant  makes  the  angle  (a)t+  P0OP,) 
=(o)  t  -f  00)  with  the  line  of  reference.  Hence  the  resultant  e.  m.  f. 
is 

Si  +  ez  =  EO  si™  (to  t  +  ^o) 
where 


334  ELECTRICAL  ENGINEERING 


E0  =V  E2l+E\+2E1E2  cos  &  (19a) 


which  are  identical  with  equations  (18). 

The  above  discussion  of  course  applies  to  any  two  harmonic 
functions  of  the  same  variable.  For  example,  the  resultant  of 
two  harmonic  currents 

ij  =/!  sin  cot 

and  i2  =/2  sin  (cot  +  0} 

is  fc\  +  i2  =I0  sin  (out  +  00) 

where 

(20o) 

(206) 


_jF"    /2  SI'TI  (/ 
aU  Ul  +  Izcos0\ 


A  re-reading  of  Articles  8  and  9  will  be  found  helpful  in 
understanding  the  vector  method  of  representing  alternating  cur- 
rents and  potential  differences. 

187.  The   Vectors   Representing   Any   Number   of   Harmonic 
Currents  and  P.D.'s  of  the  Same  Frequency  are  Stationary  with 
Respect  to  One  Another.  —  This  is  immediately  evident  from  the 
fact  that  the  analytical  expression  for  any  harmonic  function  of 
the  time  is  A  sin  (a)t  +  0),  where  A  and  0  are  constants  for  each 
such  function,  and  CD,  which  is  equal  to  277  times  the  frequency, 
is   the  angular  velocity  at  which  the  vector  representing   this 
function  rotates.     Hence  the  vectors  representing  any  number 
of  such  functions  of  the  same  frequency  all  rotate  through  equal 
angles  in  any  interval  of  time,  i.e.,  their  relative  positions  with 
respect  to  each  other  remain  unchanged.     Consequently  in  prob- 
lems which  involve  only  harmonic  currents  and  harmonic  p.d.'s 
and  their  phase  displacements  with  respect  to  each  other,  the 
vectors  representing  the  currents  and  p.d.'s  may  be  considered 
as  stationary. 

It  should  be  carefully  noted  that  non-harmonic  currents  or 
p.d.'s,  or  currents  or  p.d.'s  of  different  frequencies  cannot  be 
represented  on  the  same  diagram  by  stationary  vectors. 

188.  The    Lengths    of    the    Vectors    Representing    Harmonic 
Functions  Taken  Equal  to  their  Effective  Values.  —  In  any  prob- 
lem which  has  to  do  only  with  the  effective  values  of  harmonic 


ALTERNATING  CURRENTS 


335 


currents,  p.d.'s  and  e.m.f.'s  and  their  phase  relations,  the  lengths 
of  the  vectors  representing  these  quantities  may  be  taken  equal 
to  their  effective  values,  since  the  effective  values  are  directly 
proportional  to  the  maximum  values  of  these  quantities.  We 
then  have  that 

1.  The   effective   value   of   the   resultant   of   any   number   of 
harmonic  currents  (or  p.d.'s  or  e.m.f.'s)  of  the  same  frequency  is 
equal  to  the  vector  sum  of  the  vectors  representing  these  currents 
(or  p.d.'s  or  e.m.f.'s.) 

2.  The  average  power  corresponding  to  any  harmonic  current 
and  p.d.  of  the  same  frequency  is  equal  to  the  product  of  the 
lengths  of  the  vectors  representing  them  times  the  cosine  of  the 
angle  between  these  vectors.     See  equation  (8). 

189.  Potential  Drop  due  to  a  Harmonic  Current  in  a  Circuit 
of  Constant  Resistance  and  Inductance.  —  Let  r  be  the  resistance 


i=I0sin   (2  IT  ft) 


rl  I 

Fig.  101. 

and  L  the  inductance  between  the  two  points  1  and  2  of  the  circuit 
(see  Fig.  101),  and  let  this  circuit  be  perfectly  insulated  and  have 
neither  capacity  nor  mutual  inductance  with  respect  to  any  other 
circuit.  Let  the  current  be  i=I0  sin  (27rft),  where  70  is  the  maxi- 
mum value  of  the  current  ( =\/2  X  the  effective  value)  and  /  is 
the  frequency ;  let  v  be  the  value  of  the  instantaneous  potential 
drop  from  1  to  2  in  the  direction  of  the  current  i.  The  instantaneous 
drop  of  potential  in  the  direction  of  the  current  is  equal  to  the 
resistance  drop  in  the  circuit  plus  the  back  e.  m.  f.  in  the  coil ; 
compare  with  equation  (18d)  of  chapter  III.  The  resistance  drop 


336  ELECTRICAL  ENGINEERING  , 

is  ri  and  the  back  e.  m.  /.  (due  to  self  inductance)  is  L— (see  Article 

dt 

116).     Hence  the  total  instantaneous  potential  drop  is 

Tdi 

v=n-\-L— 
dt 

Substituting  for  i  its  value  70  sin  (2ir#)  in  the  above  equa- 
tion we  get 

v  =rI0sin  (2  TT  ft)  +  (2  TT  fL)  I0cos  (2  TT  ft) 

=rI0sin  (2  TT  ft)  +  (2  TT  /L)  70sw  (2  77  //  +  ~ 

Hence  when  the  resistance  and  inductance  are  constant  the  p.d. 
from  1  to  2  is  also  a  harmonic  function  of  the  time,  has  the  same 
frequency,  and  consists  of  two  components  having  respectively 

the  effective  values  ri  and  (2  irfL)  7,  where  7=  -4=.  is  the  effective 

V2 

value  of  the  current. 

The  first  component  of  the  p.d.,  namely  ri,  is  in  phase  with 
the  current  and  the  second  component  of  ihep.d.,  namely  (2irfL)I, 
leads  the  current  by  90°.  Hence  the  effective  value  of  the 
resultant  p.d.  is  (see  the  vector  diagram  and  also  equations  1 9) 


or  7=71r2+(27r/L)2  (21) 

and  this  resultant  p.d.  leads  the  current  by  the  angle 

0-ton-1     27rL  (21a) 


This  angle  9  is  the  power-factor  angle  of  the  circuit.     The  power 
factor  of  the  circuit  is  then 

cos  0=—=r  (216) 


Note  that  the  potential  drop  in  a  given  portion  of  a  circuit  is 
equal  to  the  e.  m.  /.  impressed  across  the  terminals  of  this  portion 
of  the  circuit,  which  e.  m.  f.  is  in  the  same  direction  around  the 
closed  circuit  as  the  potential  drop  through  the  given  portion  of 
the  circuit  ;  hence  in  the  above  formulas  V  may  be  taken  to  rep- 
resent either  the  potential  drop  in  the  circuit  or  the  e.  m.  /.  im- 
pressed on  it. 


ALTERNATING  CURRENTS  337 

Since  the  first  component  of  the  p.d.  is  in  phase  with  the 
current,  and  the  second  component  of  the  p.d.  is  in  quadrature 
with  the  current,  it  is  the  first  component  alone  which  determines 
the  average  power  put  into  the  circuit ;  rl  is  therefore  the  power 
component  of  the  p.d.  The  electric  energy  r&dt  which  is  con- 
verted into  heat  energy  in  the  resistance  r  during  each  interval 
of  time  dt  is  not  returned  when  the  current  reverses,  since  this 
energy  is  proportional  to  the  square  of  the  current  i  and  is  there- 
fore independent  of  the  direction  of  i.  The  average  rate  at  which 
this  energy  is  supplied  to  the  circuit  is  represented  by  the  constant 
term  in  the  expression  for  instantaneous  power,  equation  (8). 

As  just  noted,  the  second  component  of  v,  namely  (27T/L)/, 
is  in  quadrature  with  the  current  and  therefore  contributes  noth- 
ing to  the  average  power,  i.e.,  this  is  the  reactive  component 
of  the  p.d.  This  is  also  evident  from  the  fact  that  the  energy 
stored  in  the  magnetic  field  while  the  current  is  rising  from  zero 
to  a  maximum  is  equal  to  the  work  done  on  the  current  by  the 
field  when  the  current  decreases  from  its  maximum  value  to  zero. 
Also  note  that  this  cyclic  transfer  of  energy  to  and  from  the 
magnetic  field  occurs  twice  during  each  cycle  of  the  current,  i.e., 
it  is  represented  by  the  double  frequency  term  in  the  expression 
for  instantaneous  power,  equation  (8). 

190.  Effective  Resistance,  Reactance  and  Impedance  of  an 
Alternating  Current  Circuit.  —  In  general  when  an  alternating 
current  is  established  in  an  electric  circuit,  secondary  currents  are 
established  in  the  surrounding  conductors  (for  example,  as  "  eddy  " 
currents  in  the  iron  cores  of  the  magnetic  circuits  of  electric  ma- 
chinery) ;  a  certain  amount  of  energy  in  addition  to  that  dissipated 
in  the  main  or  primary  circuit  is  therefore  also  dissipated  in  the 
conductors  in  which  these  secondary  currents  are  established. 
Also,  due  to  the  fact  that  the  number  of  lines  of  magnetic  induc- 
tion linking  the  center  of  a  wire  is  greater  than  the  number  linking 
the  outside  of  the  wire  (see  Article  121)  the  induced  back  electro- 
motivej-inside  the  wire  is  in  general  greater  than  that  induced  in 
the  outside  filaments  of  the  wire,  and  therefore  the  current  density 
is  not  uniform  over  the  cross  section  of  the  wire,  as  is  the  case 
with  a  continuous  current;  hence  the  effective  resistance  of  a 
conductor  to  an  alternating  current  is  greater  than  its  resistance 
to  a  continuous  current.  This  skin  effect,  however,  is  not  as  a 
rule  serious  for  the  frequencies  'and  sizes  of  conductors  used  in 


338  ELECTRICAL  ENGINEERING 

practice,  except  in  the  case  of  steel  rails  used  for  conductors  in 
railway  work.  Also,  when  there  is  iron  in  the  magnetic  field 
established  by  the  current,  a  certain  amount  of  energy  is  dissi- 
pated in  the  iron,  due  to.  hysteresis.  Hence  the  average  rate 
at  which  heat  energy  is  dissipated  when  an  alternating  current 
is  established  in  a  given,  portion  of  a  circuit  is  not  equal  to  the 
product  of  the  square  of  the  effective  value  of  the  current  by  the 
resistance  of  the  conductor  in  which  the  current  is  established, 
as  determined  by  a  continuous  current  measurement,  but  is  in 
general  greater  than  this.  This  portion  of  the  circuit  may,  how- 
ever, be  considered  as  having  an  effective  or  apparent  resistance  r 
such  that  this  resistance  multiplied  by  the  square  of  the  effective  value 
of  the  current  gives  the  true  average  rate  at  which  heat  energy  is  dis- 
sipated when  the  current  is  established  in  this  portion  of  the  circuit. 
This  effective  resistance  is  in  most  practical  cases  approximately 
independent  of  the  current  strength  (as  measured  by  the  effective 
value)  but  does  depend  upon  the  frequency  of  the  current,  and, 
in  case  there  is  a  loss  of  energy  due  to  hysteresis,  upon  the  effective 
value  and  the  wave  shape  of  the  current  also.  (The  part  of  the 
effective  resistance  which  takes  into  account  the  hysteresis  loss 
in  iron  is  not  strictly  constant,  but  varies  approximately  as  the 
1.6  power  of  the  maximum  flux  density,  and  therefore  as  the 
1 .6  power  of  the  maximum  value  of  the  current.) 

The  average  rate  at  which  heat  energy  is  dissipated  when  an 
alternating  current  is  established  in  any  portion  of  a  circuit  may 
then  be  written 

Ph  =  rP  (22) 

where  r  is  the  effective  resistance  of  this  portion  of  the  circuit 
and  7  is  the  effective  value  of  the  current. 

The  effective  value  of  the  resultant  potential  drop  in  any 
portion  of  a  circuit,  however,  is  in  general  greater  than  rl.  Let 
V  be  the  vector  representing  the  resultant  p.d.  in  any  portion  of 
the  circuit  and  let  E  be  the  vector  representing  any  externally  in- 
duced e.  m.  /.  in  this  portion  of  the  circuit,  both  in  the  direction 
of  the  current  (e.g.,  if  the  portion  of  the  circuit  considered  is  the 
armature  of  an  alternator,  E  is  the  induced  e.  m.  f.  due  to  the 
relative  motion  of  the  armature  and  the  magnetic  field).  Then 
the  ratio  of  the  numerical  value  of  the  vector  difference  E  —  V 
to  the  effective  value  of  the  current  I  in  this  portion  of  the  circuit  is 


ALTERNATING  CURRENTS  339 

defined  as  the  impedance  of  this  portion  of  the  circuit.     That  is, 
the  impedance  of  this  portion  of  the  circuit  is 

E-  V 
Z=—J-  (23) 

When  there  is  no  externally  induced  e.  m.  /.,  that  is,  when  E  is 
zero,  the  impedance  of  the  given  portion  of  the  circuit  is 

V 
z=j  (23a) 

The  square  root  of  the  difference  between  the  square  of  the 
impedance  of  any  portion  of  a  circuit  and  the  square  of  the  effective 
resistance  of  this  portion  of  the  circuit  is  called  the  reactance  of 
this  portion  of  the  circuit,  and  is  usually  represented  by  the 
symbol  x.  The  reactance  corresponding  to  the  impedance  z 
and  the  effective  resistance  r  is  then 


r2  (24) 

In  the  case  o£  a  harmonic  current  in  a  given  portion  of  cir- 
cuit in  which  there  is  no  externally  induced  electromotive  force, 
rl  is  the  component  of  the  p.d.  in  phase  with  the  current,  and 
xl  is  the  component  of  the  p.d.  in  quadrature  with  the  current. 
When  the  current  lags  behind  the  potential  drop  in  the  direction 
of  the  current  the  reactance  is  taken  as  positive,  when  the  current 
leads  the  p.d.  the  reactance  is  taken  as  negative;  that  is,  the  sign 
of  x  is  chosen  so  that  xl  represents  the  component  of  the  p.d.  90° 
ahead  of  the  current.  The  angle  by  which  the  current  lags  behind 
the  p.d.,  or  by  which  the  p.d.  leads  the  current,  is  then 

0=tan-L-  (25) 

r 

and  the  power  factor  of  the  circuit  is 

cos  0  =       T    =-  =-  (25a) 


It  also  follows  from  the  definition  of  reactive  power    (Article 
184),  that  the  reactive  power  is  equal  to  xP. 

Since  both  reactance  and  impedance  are  ratios  of  p.d.  to 
current  they  are  both  measured  in  the  same  unit  as  resistance, 
i.e.,  in  practical  units  both  the  reactance  and  the  impedance  are 
expressed  in  ohms. 


340  ELECTRICAL  ENGINEERING 

191.  Reactance  and  Impedance  of  a  Coil  of  Constant  Resist- 
ance and  Inductance*  to  a  Harmonic  Current.  —  From  equations 
(21)  and  the  above  definitions,  it  is  evident  that  the  impedance 
of  a  circuit  of  constant  resistance  (independent  of  the  value  of 
the  current  at  any  instant)  and  a  constant  inductance  (inde- 
pendent of  the  value  of  the  current  at  each  instant),  to  a  har- 
monic current  is 


/L)2  (26) 

and  the  reactance  is 

^=27T/L  (26a) 

Note  that  these  relations  are  deduced  on  the  assumptions  that 
the  coil  has  no  electrostatic  capacity  and  that  the  current  is  a 
sine  wave. 

192.  Reactance  and  Impedance  of  a  Coil  of  Constant  Resist- 
ance and  Inductance  to  a  Non-Harmonic  Current.  —  The  reactance 
of  such  a  circuit  to  a  non-harmonic  current  is  readily  found 
when  the  equation  of  the  current  is  known.  Consider  the  special 
case  of  a  current  which  contains  the  third  harmonic.  This 
current  may  be  represented  by  the  equation 

i  =/j  sin  (2  77  ft)  +  Is  sin  (6  77  ft  +  6) 

where  A  and  73  are  the  maximum  values  of  the  fundamental  and 
third  harmonic  respectively.  When  the  wave  shape  remains 

constant  for  all  values  of   the   current  —  is  a  constant.     Then 

/, 

the  instantaneous  value  of  the  drop  of  potential  through  the 
resistance  and  inductance  is 

di 

v  =ri  +  L—  =rll  sin  (2  TT  ft)  +  rI3  sin  (6  77  ft  +  0) 
dt 

+  (2  TT  fL)  I,  COS  (2  77  ft)  +  (6  77  fL)  73  COS  (6  TTft+0) 

Combining  the  terms  of  the  same  frequency  in  the  manner  de- 
scribed in  Article  185,  we  get 


v  =  vV  +  (2  77  fL)2 1 1  sin  (2  77  ft  +  a,) 
+  vV2+(677/L)2  ^3  sin  (6  77  ft  +  a2) 
where 

*  Such  a  coil  is  frequently  called  an  "  impedance  "  coil. 


ALTERNATING  CURRENTS  341 

a^tan-1  2-^lL  and  a^O  +  tan1  67r/L 
r  r 

(see  equation  186). 

The  effective  value  of  V  is  then  (see  Article  176) 


and  the  effective  value  of  i  is 


whence 


7  7 2  7 2 

Since  —  is  constant,  — — — -  is  constant;  put  a2= .       Then 

*i  A  ~t~73  7X  +73 

the  impedance  of  the  circuit  to  this  current  is  equal  to 


*  =  V  r2  +  (2  TT  /L)2  (1  +  8a2)  (27) 

and  the  reactance  is 


x  =\/z2-r>  =2"nr  /TV  i  +  8a2  (27a) 


The  constant  a  is  the  ratio  of  the  effective  value  of  the  third 
harmonic  to  the  effective  value  of  the  resultant  current. 

Note  that  the  effective  value  of  the  first  harmonic  in  the  p.d. 

I,       '  _ 
wave  is—  V  r*  +  (2  7T/L)2,  while  the  effective  value  of  the  third 


73 


harmonic  is  — =  V  r2  +  (6  TT  /L)2.    When  the  inductance  L  is  large 
\  & 

compared  with  the  resistance,  the  ratio  of  the  third  harmonic 
in  the  p.d.  wave  to  its  fundamental  must  then  be  three  times 
as  great  as  the  ratio  of  the  third  harmonic  to  the  fundamental 
in  the  current  wave.  Vice  versa,  the  third  harmonic  in  the 
current  wave  resulting  from  a  given  p.d.  will  be  relatively  only 
one  third  as  great  in  the  current  wave  as  in  the  p.d.  wave.  There- 
fore an  inductance  in  a  circuit  tends  to  dampen  out  the  harmonics 
in  the  current  wave  when  a  non-harmonic  e.  m.  f.  is  impressed  on 
it,  and  to  make  this  wave  approach  more  nearly  to  a  sine  wave. 
The  higher  the  order  of  the  harmonic  the  greater  the  damping, 


342  ELECTRICAL  ENGINEERING 

193.  Current  through  a  Condenser  when  a  Harmonic  P.D.  is 
Established  across  It.  —  Capacity  Reactance.  —  Let  the  p.d.  across 
the  terminals  of  the  condenser  be 

v  =  V0  sin  2  TT  ft 


and  let  the  total  current  through  the 
condenser  in  the  direction  of  the  po- 
tential drop  be  i.  Let  C  be  the  capacity 
of  the  condenser  and  g  the  conductance 
of  the  dielectric  between  its  plates,  i.e., 
the  leakance  of  the  condenser.  Then, 
since  the  total  current  through  the  con- 
denser is  the  sum  of  the  conduction  cur- 
rent, gv,  through  the  dielectric  plus  the 

displacement  current  C —    through  the" 
dt 

dielectric  (see  Article  151),  we  have 


Fig.  102.  dt 

Therefore 

i=gV0  sin  (2  TT  ft)  +  (2  TT  fC)  V0  cos  (2  TT  ft) 

=g  V0  sin  (2  TT  ft)  +  (2  TT  fC)  V0  sin  (2Trft  +  -) 

Hence  the  component  of  the  current  through  the  condenser  due 
to  the  conductance  of  the  dielectric,  i.e.,  the  leakage  current, 
is  in  phase  with  the  p.d.  across  the  condenser,  and  the  component 
of  the  current  through  the  condenser  due  to  its  capacity,  i.e., 
the  displacement  or  "  charging  "  current,  leads  the  p.d.  by  90°. 
The  effective  value  7  of  the  total  current  is 


where  V  is  the  effective  value  of  the  p.d.  Hence  the  effective 
resistance  of  a  leaky  condenser  to  an  alternating  current  of  fre- 
quency /  is  (see  Article  190) 

(28) 


P        <f+(27T/C)2 

and  its  impedance  is 

zc  =—  = —       l  (28a) 

/ 
and  therefore  its  reactance  is 


ALTERNATING  CURRENTS  343 


F       (286) 

xc  is  negative,  since  the  current  leads  the  p.d.  When  the  dielec- 
tric is  a  perfect  insulator  #=0,  and  in  this  case  rc=0  and  the 
reactance  of  the  condenser  reduces  to 


The  reactance  —  -  corresponding  to  a  capacity  alone,  with- 

2  77  fC 

out  leakance,  is  called  the  capacity  reactance  of  the  condenser; 
it  must  not  be  confused  with  the  effective  reactance  of  the  con- 
denser; the  .two  are  equal  only  when  there  is  no  leakage.  The 
leakage  conductance  of  condensers  used  in  practice  is  usually 
.quite  small  but  is  not  always  negligible. 

194.  Current  through  a  Condenser  when  a  Non-Harmonic 
P.D.  is  Established  across  It.  —  When  the  p.d.  contains  the  third 
harmonic,  for  example,  its  equation  is 

v  =  V,  sin  (2  TT  ft)  +  V3  sin  (Qirft  +  0). 
Neglecting  the  leakance  of  the  condenser,  the  current  is  then 

«,c* 

dt 

=-•(2  7T  fC)   V,  COS  2  7T  #+  (6  7T/C)  V3  COS  (6irft  +  0) 

The  effective  value  of  the  p.d.  is,  see  Article  176, 


2         2 
and  the  effective  value  of  the  current  is,  see  Article  176, 


(29) 


where  a  =  ^1          3        =  ratio  of  the  effective  value  of  the  third 

'    1       I      V  3 

harmonic  of  the  p.d.  to  the  resultant  p.d. 

When  the  leakance  of  the  condenser  is  negligible  the  ratio  of 
the  third  harmonic  to  the  fundamental  in  the  current  wave  is 
therefore  three  times  as  great  as  the  ratio  of  the  third  harmonic 
to  the  fundamental  in  the  p.d.  wave.  Hence,  when  a  non- 
harmonic  p.d.  is  impressed  across  a  condenser,  the  upper  har- 
monics in  the  current  wave  are  greater  than  the  corresponding 


344 


ELECTRICAL  ENGINEERING 


harmonics  in  the  p.d.  wave  directly  in  proportion  to  their  order. 
For  example,  when  the  seventh  harmonic  in  the  p.d.  wave  has 
an  amplitude  equal  to  5  per  cent  of  the  fundamental,  the  seventh 
harmonic  in  the  current  wave  has  aa  amplitude  equal  to  35  per 
cent  of  the  amplitude  of  the  fundamental  in  the  current  wave. 
Compare  with  the  effect  produced  by  an  inductance,  Article  192. 
195.  Impedance  of  a  Resistance,  Inductance  and  Capacity  in 
Series  to  a  Harmonic  Current.  —  When  a  harmonic  current  of 
effective  value  7  is  established  in  such  a  circuit  (see  Fig.  103),  the 
p.d.  across  the  resistance  is  Vr  =rl  and  is  in  phase  with  7.  The  p.d. 
across  the  inductance  is  VL  =  (2  IT  fL)  I  and  leads  I  by  90°.  The 
p.d.  across  the  capacity  (a  condenser  with  negligible  leakance)  is 
7 


27T/C 


-  and  lags  behind  7  by  90°.     Hence  the  resultant  p.d.  is 


27T/C 


(30) 


=  I0  sin 


(SirfL)I/ 


I      v 


Fig.  103. 

The  impedance  of  such  a  circuit  is  therefore 

v    iT 

z=~f=\  r 
The  reactance  is  therefore 


2ir/C/ 


The  angle  by  which  the  p.d.  leads  the  current  is 


(30a) 
(306) 
(30c) 


ALTERNATING  CURRENTS  345 

Note  that  equations  (30)  apply  only  in  case  the  condenser  has 
no  leakance. 

196.  Resonance.  —  When  the   inductance  L  and  capacity  C 
in  the  case  just  considered  are  of  such  values  that 

27T/L=- 

orwhen  27rfC 


the  reactance  of  the  circuit  is  zero  and  the  impedance  is  equal  to 
the  resistance,  and  therefore  the  current  corresponding  to  a 

y 
given    p.d.  V  is  /=—  ;  that  is,  the  current  is  a  maximum  and 

depends  only  upon  the  resistance  of  the  circuit.  The  frequency 
corresponding  to  this  condition  is  the  same  as  the  frequency 
with  which  the  current  and  p.d.  would  oscillate  were  the  con- 
denser short-circuited  by  the  inductance;  i.e.,  this  frequency 
corresponds  to  the  free  period  of  such  a  circuit.  (See  Article 
164.)  Note  the  analogy  with  the  motion  of  a  body,  which  is  free 
to  vibrate,  produced  by  a  periodic  force  having  a  period  equal  to 
the  period  of  vibration  of  the  body  ;  for  example,  a  heavy  church 
bell  may  be  caused  to  swing  with  large  amplitude  (corresponding 
to  the  maximum  value  of  the  current  /)  when  a  comparatively 
small  force  (corresponding  to  the  p.d.  V)  is  applied  by  the  man 
pulling  the  rope,  provided  the  successive  applications  of  this 
force  are  in  time  with  the  swinging  of  the  bell. 

When  the  frequency  of  the  electromotive  force  impressed 
across  the  terminals  of  the  circuit  is  equal  to  the  natural  or  free 
frequency  of  the  circuit,  the  circuit  is  said  to  be  in  resonance 
with  this  impressed  electromotive  force. 

Note  also  that,  although  the  resultant  p.d.  across  the  resist- 
ance, inductance  and  capacity  in  series,  is  equal  to  rl,  i.e.,  is  the 
same  as  the  p.d.  across  the  resistance,  the  p.d.  across  the  induc- 
tance or  across  the  condenser  may  be  many  times  this.  For 
example,  when  the  inductance  L  is  1  henry  and  the  capacity  C 
is  7.04  microfarads,  and  the  frequency  is  60  cycles,  the  inductive 
reactance  is  XL  =2  TT  X  60  X  1  =377,  and  the  capacity  reactance  is 

xr  —  —  -  —  =  —  377.     Hence  the   total  reactance 

27TX60X7.04X10-6 


346  ELECTRICAL  ENGINEERING 


of  the  circuit  is  xL  +  xc=Q,  and  the  circuit  is  in  resonance  with  the 
impressed  e.  m.  /.  When  the  resistance  r  is  1  ohm  and  the  im- 
pressed e.  m.  f.  across  the  entire  circuit  is  100  volts,  the  current  is 

100 
/  =    ,  =100  amperes 

V  (\Y+(W 

The  p.d,  across  the  resistance  is  then  100X1=100  volts.  The 
p.d.  across  the  inductance,  however,  is  2  TT  X60X1  X  100  =37,700 
volts  and  the  p.d.  across  the  condenser  is 

inn 

_  —  -  =37,700  volts 
27TX60X7.04X10-6 

This  brings  out  in  a  striking  manner  the  fundamental  fact  that 
alternating  p.d.'s  cannot  be  added  algebraically;  they  must  be  added 
vectorially. 

197.  Impedances  in  Series.  —  In  the  case  of  continuous  cur- 
rents we  have  seen  that  several  resistances  r1;  r2,  r3,  etc.,  in  series 
are  equivalent  to  a  single  resistance  R  which  is  equal  to  the  arith- 
metical sum  of  7*1,  r2,  r3,  etc.,  that  is  R=rl  +  r2  +  r3-\  ---  .  In 
case  of  alternating  currents  this  same  relation  -also  holds  for  the 
resistance  of  any  number  of  impedances  in  series,  such  as  a  num- 
•ber  of  coils  of  wire  in  series.  For,  when  the  same  current  / 
flows  through  each  coil  the  average  power  dissipated  in  all  the 
coils  is  r1/2  +  r272  +  r3/2  +  etc.  =(r1  +  r2  +  r3  +  etc.)/2  where  rl}  r2,  r3, 
etc.,  are  the  effective  resistances  of  the  respective  coils.  Hence, 
calling  R  the  effective  resistance  of  all  the  coils  in  series,  we  have 
from  Article  190  that 

JR=r1  +  r2  +  r3  +  etc.  (32a) 

In  the  case  of  a  harmonic  current  the  total  potential  drop  through 
all  the  coils  in  phasewith  the  current  is  then  RI  =(rt  +  r2  +  r3  +  etc.)/. 

Similarly,  calling  x^  x2,  x3,  etc.,  the  reactances  of  the  respec- 
tive coils  to  the  harmonic  current  /,  the  p.d.'s  across  the  separate 
reactances  are  respectively  xj,  xj,  xal,  etc.,  and  are  all  in  quad- 
rature with  the  current  and  are  therefore  either  in  the  same  or  op- 
posite direction.  Hence  the  algebraic  sum  of  these  p.d.'s  gives  the 
total  p.d.  in  quadrature  ahead  of  the  current.  Whence,  calling 
X  the  equivalent  reactance  of  the  circuit,  we  have  from  Article 
190  that 

XI  =xj  +  xj  +  x3I  +  etc. 
or 

X  =xl  +  x2  +  x3  +  etc.  (326) 


ALTERNATING  CURRENTS 


347 


(Note  that  the  reactances  x  may  be  either  positive  or  negative; 

an  inductive  reactance  is  positive,  a  capacity  reactance  negative.) 

The  resultant  p.d.  across  all  the  coils  in  series  is  therefore 


or 


Whence,    the  equivalent   impedance    of   such  a  circuit  is,  from 
Article  190, 

V 
=7 


etc.)2    (32c) 


Hence,  the  equivalent  impedance  of  any  number  of  impedances 
zif  z2,  za,  etc.,  is  not  the  sum  of  the  separate  impedances.  In 
general,  the  equivalent  impedance  can  be  calculated  only  when  the 
resistance  r  and  the  reactance  x  of  each  impedance  is  known. 

Example:  An  alternating  current  of  100  amperes  is  to  be 
supplied  to  a  receiver  which  has  an  equivalent  resistance  rl  of  2 
ohms  and  an  equivalent  reactance  xl  of  0.5  ohm.  The  line 
has  a  resistance  r2  of  0.1  ohm  and  an  inductive  reactance  x2  of  1.5 
ohms.  The  equivalent  resistance  of  the  line  and  receiver  is  then 
R—2  +  0.1  ==2.1  ohms  and  the  equivalent  reactance  of  the  line 
and  receiver  is  X  =0.5  +  1.5  =2.0  ohms.  Hence  the  equivalent 
impedance  of  the  line  and  receiver  is  Z=  1/(2.1)2  +  (2.0)2=2.90 
ohms.  The  impedance  of  the  receiver  alone  is  21=J//(2)24-(0.5)2 
=2.06  and  the  impedance  of  the  line  alone  is  22=1/(0.1)24-(1.5)2 
=  1.50.  Hence  2,4-22  =3.  56  which  is  23  per  cent  greater  than 
the  true  impedance  of  the  line  and  receiver. 

When  the  current  sup- 
plied to  the  receiver  is  100 
amperes,  the  p.d.  at  the  re- 
ceiver is  V  =100X2!  =100 
X  2.  06  =206  volts  arid  the 
p.d.  at  the  generator  is 
V0=  100  XZ  =  100  X  2.90 
=  290  volts  ;  that  is,  the  p.d. 
at  the  receiver  is  290-206 
=  84  volts  less  than  at  the  TL!~  200 

generator.     The  total  poten-  Fig.  104. 

tial  drop   in  the  two   wires  forming   the   line,   however,   is   100 


348  ELECTRICAL  ENGINEERING 


=  100  X  1.50=150  volts,  which  is  79  per  cent  greater  than 
the  true  difference  between  the  potential  drops  across  the  gen- 
erator and  across  the  receiver"  terminals. 

Fig.  104  will  make  these  relations  clear.  The  reason  the  p.d. 
in  the  line  is  not  equal  to  the  difference  of  the  p.d.'s  at  the 
generator  and  the  receiver  is  that  the  p.d.  in  the  line  and  the 
p.d.  at  the  receiver  aretnot  in  phase. 

198.  Impedances  in  Parallel.  —  In  the  case  of  continuous  cur- 
rents we  have  seen  (Article  98)  that  when  several  resistances 
ri,  1*2,  r3,  etc.,  are  connected  in  parallel,  the  currents  in  the  various 

V  V          V 

resistances  are  respectively/!  =—  ,  /2  =—  ,  /3  =—  ,  etc.,  and  therefore 

r,  r2          r3 

that  the  total  current  between  the  junction  points  of  the  several 

resistances  is  7=/1  +  /2  +  /3+  etc.,  =  V  (_  +  _+_H  ---  ). 

XT,     r2     r3  / 

Whence  the  equivalent  resistance  R  must  be  such  that 

J  ^=1  +  1+1  +  --- 
R      V     r,      r2     r, 

In  the  case  of  alternating  currents,  the  currents  in  any  number 


Fig.  105 

of  impedances  connected  in  parallel  between  two  points  A  and  B 
(Fig.  105)  are  respectively 

_V 


V 
/.=—   etc. 

z3 

but  these  currents  are  not  in  phase,  hence  they  cannot  be  added 


ALTERNATING  CURRENTS  349 

algebraically ;  they  must  be  added  vectorially.  The  components 
of  the  various  currents  in  phase  with  the  potential  drop  V  between 
A  and  B  are  respectively 

r         a      V  rl      rl  ,. 
A  cos  ul  = =— V 

zl  zl      z? 

72 cos  02=--2=i Fete. 


and  the  components  of  these  currents  in  quadrature  behind  V  are 
respectively     . 

7X  sin  #,=--=— V 

z,  z,     z? 

72  sin  02=-9-=?LV  etc. 
The  total  current  in  phase  with  V  is  then 

and  the  total  current  in  quadrature  behind  V  is 


Hence  the  total  current  /  is  the  vector  sum  of  these  two  com 
ponents,  i.e., 


and  therefore  the  equivalent  impedance  is  Z,  where 

z=v=\  b?4"^"  ~J  +b?+3+~  ~     (33) 

199.  Conductance,  Susceptance  and  Admittance.  —  Note  that 

7* 

for  any  of  the  impedances  the  factor  —  multiplied  by  the  p.d. 
across  the  impedance  gives  the  component  of  the  current  in  phase 
with  the  p.d.  and  therefore  this  factor  —  multiplied  by  the  square 

rV2 
of  the  p.d.,  i.e.,    —    gives  the  average  power  dissipated  in  this 

z2 

impedance.  In  general,  the  ratio  of  the  average  rate  Ph  at  which 
heat  energy  is  dissipated  in  any  part  of  a  circuit  to  the  square  of 
the  effective  value  of  the  p.d.  V  across  this  part  of  the  circuit,  is 


350  ELECTRICAL  ENGINEERING 

called  the  effective  conductance  "  g"  of  this  part  of  the  circuit, 

that  is, 

-p 

g==-yi  (34) 

The  ratio  of  the  effective  value  of  the  current  in  any  part  of  a 
circuit  to  the  numerical  value  of  the  vector  difference  of  the  ex- 
ternally induced  e.  m.  f.  E  and  the  resultant  p.d.  V  in  this  portion 
of  the  circuit  is  called  the  admittance  "  y  "  of  this  part  of  the 
circuit,  that  is, 

/ 

~E^V  (35) 

The  square  root  of  the  difference  between  the  square  of  the  ad- 
mittance of  any  part  of  a  circuit  and  the  square  of  the  effective 
conductance  of  this  part  of  the  circuit  is  called  the  susceptance 
"  b  "  of  this  part  of  the  circuit,  that  is, 

&=Vy-<f  (36) 

In  the  case  of  a  harmonic  p.d.,  established  across  the  terminals 
of  a  circuit  in  which  there  is  no  externally  induced  e.  m.  /.,  the 
product  of  the  susceptance  and  the  p.d.  gives  the  component  of 
the  current  in  quadrature  with  the  p.d.,  since  the  total  current 
is  yV  and  the  component  in  phase  with  V  is  gV,  whence  the 


component  in  quadrature  with  V  is  \/(?/V)2—  (gV)2  = 
The  sign  of  the  susceptance  b  is  taken  positive  when  the  p.d. 
leads  the  current,  negative  when  the  p.d.  lags  behind  the  current  ; 
that  is,  the  susceptance  and  reactance  of  a  circuit  always  have 
the  same  sign.  Also,  from  the  definition  of  reactive  power  (Article 
184),  it  follows  that  the  reactive  power  is  bV2. 

From  the  above  definitions  and  the  discussion  in  Article  193, 
it  follows  that  the  effective  conductance,  or  as  it  is  also  called, 
the  leakance,  of  a  condenser,  is  equal  to  the  reciprocal  of  the 
resistance  of  the  dielectric  between  its  plates,  as  measured  by 
means  of  a  continuous  current  (provided  there  is  no  dissipation 
of  energy  due  to  "  dielectric  hysteresis  ")  and  the  susceptance  of 
a  condenser  of  capacity  C  to  a  harmonic  current  of  frequency  /  is 

bc  =  -2>rrfC. 

The  susceptance  of  a  condenser  is  negative,  since  the  charging 
current  corresponding  to  the  capacity  C  leads  the  p.d.  The  ad- 
mittance of  a  condenser  having  a  leakance  g  and  capacity  C  to  a 
harmonic  current  of  frequency  /  is  then 


ALTERNATING  CURRENTS  351 


Compare  with  the  resistance,  reactance  and  impedance  of  a  coil. 

Also,  from  the  above   definitions,  we  have  in  general  that, 

corresponding  to  an  impedance  z  =  i/r2  +  x*}  the   conductance  is 

r 

g=z2  (37a) 

the  admittance  is 

1 

y=-  =  Vg2  +  b2  (376) 

z 

and  the  susceptance  is 

=z2  (37c) 

Hence  the  resultant  admittance  of  any  number  of   circuits  in 
parallel,  equation  (33),  is 

-62  +  63--)2  (37d) 


where  glt  g2,  g3,  etc.,  and  6t,  62,  b3,  etc.,  are  the  conductances  and 
susceptances  respectively  of  the  various  circuits. 

Since  conductance,  susceptance  and  admittance  are  the  ratios 
of  current  to  voltage,  the  unit  in  which  these  quantities  are  meas- 
ured is  the  reciprocal  of  the  ohm  ;  these  quantities  are  therefore 
expressed  in  "  mhos." 

For  circuits  in  series,  it  is  more  convenient  to  use  the  quantities 
resistance,  reactance  and  impedance.  For  circuits  in  parallel,  the 
conductance,  susceptance  and  admittance  are  more  convenient, 
since  conductances  and  susceptances  of  parallel  circuits  are  re- 
spectively additive.  Note  that  when  g  and  b  are  given  r  and  x 
can  be  immediately  calculated,  since 

r-l  (38) 


and 


(38a) 


Compare  with  equations  (37). 

200.  Admittance  of  an  Inductance  and  Capacity  in  Parallel  to 
a  Harmonic  Current.  —  Resonance.  —  Let  L  be  the  inductance  of 
a  coil  of  negligible  resistance,  C  the  capacity  of  a  condenser  having 


352  ELECTRICAL  ENGINEERING 

negligible  leakance,    and  /  the   frequency  of  the  current.       Let 

the  coil  and  condenser  be  connected  in  parallel  as  shown  in  Fig.  106. 

L  Then,  the  susceptance  of  the 


-    ^uuuu 

uuu^- 

coil  is  b  —            '  and  the  sus- 

B                        2wfL' 

1  '  ceptance  of  the  condenser  is 

bc  =  —  2  TT/C.    Hence  the  ad- 
mittance of  the  inductance 
and  capacity  in  parallel  is 

Fig 

c 

106. 

-27T/C 


27T/L 

Hence  the  total  current  from  A  to  B  when  a  drop  of  potential  V 
is  established  from  A  to  B  is 


277/L 

When =27Tj'C,    that    is,    when   /= the     total 

27T/L  27TI/LC 

admittance    is     zero     and    therefore    the    total     current    from 
A  to  B  is  zero.     This  is  also  evident  from   the  fact   that    the 

y 
current  in  the  inductance  is and  lags  behind  the  p.d.  by  90°; 

27T/L 

the  current  in  the  condenser  is  27T/CF  and   leads  the  p.d.  by 
90°.     Hence,  when  2  77 /C=—     -  ,  the  current  in  the  inductance 

•  2  77/C 

at  each  instant  is  exactly  equal  and  opposite  to  the  current  in 
the  condenser,   and  therefore  the  total   current   is   zero.     Note 

that  the  frequency  /= -  is  the  natural  frequency  of  the 

2771/LC 

closed   circuit  formed    by   the   condenser  and   the    inductance. 
(Compare  with  Article  196.) 

The  above  deductions  are  based  on  the  assumption  that  a 
harmonic  or  sine-wave  current  is  established  in  both  the  capacity 
and  in  the  inductance.  In  the  ideal  case  of  a  coil  of  no  resistance, 
a  harmonic  p.d.  impressed  across  its  terminals  does  not  necessarily 


ALTERNATING  CURRENTS  353 

establish  a  harmonic  current,  but  the  expression  for  the  current 
may  contain  a  constant  term.  In  the  case  of  any  actual  coil, 
however,  the  resistance  of  the  coil  (which  can  never  be  actually 
zero  though  it  may  be  quite  small)  causes  the  constant  term  in 
the  expression  for  the  current  to  become  zero  after  a  short  time, 
and  the  current  becomes  a  true  harmonic  current,  see  Article  201. 
Also,  due  to  the  resistance  of  the  coil,  and  the  leakance  of  the 
condenser  as  well,  the  resultant  admittance  of  a  coil  and  condenser 
in  parallel  can  never  be  zero  and  hence  for  a  given  p.d.  across  their 
common  terminals  the  total  current  can  never  be  absolutely  zero. 
However,  when  the  resistance  of  the  coil  and  the  leakance  of  the 
condenser  are  small  compared  with  the  reactance  of  the  coil  and 
the  susceptance  of  the  condenser,  the  total  current  will  be  a 
minimum  when  the  impressed  e.  m.  f.  has  the  frequency 

1 


Example:      When  L  —  \  henry  and  (7=7.04  microfarads,  and 

the    frequency    /   is     60    cycles   per   second,    -  .=  -  and 

2  TT/L     377 

2  7T/C  =—  and  therefore  the  total  admittance,  neglecting  the 
377 

resistance  and  leakance,  is  zero.  Hence,  when  a  coil  of  inductance 
L  and  a  condenser  of  capacity  C,  connected  in  parallel,  are 
connected  to  a  60-cycle  generator  and  a  sufficient  time  is  allowed 
for  the  current  in  both  the  condenser  and  the  coil  to  become 
true  harmonic  currents,  a  potential  drop  of  100  volts  across 
the  common  terminals  A  and  B  will  establish  a  current  of 

100 

-  =0.265  amperes  in  the  condenser  and  also  in  the  inductance, 
377 

provided  the  resistance  of  the  coil  and  the  leakance  of  the  con- 
denser are  small  compared  with  the  reactance  of  the  coil  and  the 
susceptance  of  the  condenser  respectively. 

201.  Transient  Effects  Produced  when  a  Harmonic  E.  M.  F.  is 
Impressed  on  a  Circuit.  —  So  far,  we  have  considered  the  relation 
between  current  and  p.d.  in  various  types  of  circuits  when  a  har- 
monic alternating  current  is  established  in  the  circuit.  However, 
when  a  harmonic  e.  m.  f.  is  impressed  across  the  terminals  of  a 
circuit,  time  is  required  for  a  harmonic  current  to  become  estab- 
lished just  as  in  the  case  of  a  constant  e.  m.  /.  impressed  upon  a 


354 


ELECTRICAL  ENGINEERING 


circuit  time  is  required  for  a  continuous  current  to  become  estab- 
lished. (See  Article  160.)  That  is,  when  an  alternating  e.  m.  f.  is 
impressed  across  the  terminals  of  a  circuit,  the  current  at  the 
start  is  an  oscillating  current  (see  Article  167)  and  becomes  an 
alternating  current  with  fixed  maximum  positive  and  negative 
values  only  after  the  lapse  of  an  appreciable,  though  usually 


Fig.  107. 

small,  interval  of  time.  Let  the  circuit  be  as  shown  in  Fig.  107; 
r  and  L  represent  the  resistance  and  inductance  of  an  impedance 
coil,  and  C  and  g  the  capacity  and  the  leakance  of  a  condenser 
Let  the  impressed  e.  m.  f.  be  e  =  E  sin  (27T/£  +  /8).  The  general 
differential  equations  of  this  circuit  have  already  been  given 
(Article  159)  and  are 


i=gv+C- 
dt 

v=e  —  ri—L 


eft 

dt 


(6) 


where  i  is  the  instantaneous  current  in  the  impedance  coil  (equal 
to  the  .total  displacement  and  leakage  current  of  the  condenser) 
and  v  is  the  p.d.  across  the  condenser  in  the  direction  of  the  cur- 
rent. Substituting  in  (a)  the  value  of  v  from  (b)  we  get 

dt2  dt  .  dt 

This  is  a  differential  equation  of  the  second  order.  It  is  inte- 
grated by  finding  first  the  complementary  function,  i.e.,  the  solu- 
tion corresponding  to  the  right-hand  number  equal  to  zero,  and 
then  adding  to  this  the  particular  integral.  The  solution  corre- 
sponding to  the  right-hand  number  zero  is  i  =  Aeat,  where  the 
value  of  a  is  found  by  substituting  this  value  of  i  in  the  equation 
(c),  which  gives 


ALTERNATING  CURRENTS 


355 


whence 


Put 


2\L     C/      >  4  \L     C 


LC 


. 

2\L     C 

Then  the  complementary  function  is 


J 
LC 


where  ^and  A2  are  constants  of  integration. 

The  particular  integral  of  equation  (c)  is  of  the  form 
i=I0sin(2irft  +  p-0),  where 

E 


and 


0=tanl( 


f+v- 


where  6  =  —  2  ?r/C  and  x  =2  77  /L.  That  this  is  true  can  be  seen  by 
substituting  i=I0  sin  (27rft  +  fi  —  6}  in  equation  (c),  putting  for  /0 
and  0  the  values  given  by  (/)  and  (g). 


Note  that  x  + 


represents  the  equivalent  reactance  and 


r+ 


represents  the  equivalent  resistance  of  the  entire  circuit. 


Hence  the  particular  integral  70  sin  (2  TT  ft -}- /B  —  0)  represents  a 
current  equal  to  the  impressed  e.  m.  /.  divided  by  the  equivalent 
impedance  of  the  entire  circuit  and  lags  behind  the  impressed 
e.  m.  /.  by  the  angle  corresponding  to  the -power-factor  angle  of  the 
entire  circuit.  That  is,  the  particular  integral  is  the  final  alternat- 
ing current  which  is  established  in  the  circuit  by  the  impressed 
c.  m.  f. 

The  complete  solution  of  the  current  equation  (c)  is  then 


356  ELECTRICAL  ENGINEERING 

-ut    r-          ml  -ml 


(39) 
where 


v 


„  -          -- 


4     L     C7       LC 


sn 


The  constants  of  integration  Aj  and  A2  are  determined  from 
the  values  of  the  current  i0  and  the  p.d.  v0  across  the  condenser 
at  time  t  =0. 

Since  the  constant  u  is  always  greater  than  the  constant  m, 
the  transient  term  €~ut[Al^nt  -f  A2€~mt]  in  equation  (39)  becomes 
smaller  and  smaller  as  time  increases.  In  most  practical  cases, 
only  a  fraction  of  a  second  is  necessary  for  this  term  to  become 
negligible  in  comparison  with  the  term 

in  (2  TT  ft  +  ft— tan'1  —\  representing  the  true  alternat- 
ing current.  Hence  the  transient  term  is  usually  neglected,  as  most 
problems  which  arise  in  ordinary  practical  work  have  to  deal  only 
with  the  steady  state  produced  in  a  circuit  when  a  given  e.m.f.  is 
impressed  upon  it.  However,  there  are  cases  in  which  the  tran- 
sient term  becomes  of  paramount  importance;  in  particular,  in 
predetermining  the  effect  of  switching  on  or  off  a  heavy  load 
from  a  transmission  line,  the  short-circuiting  -  or  grounding  of  a 
line,  and  the  effect  of  lightning  discharges. 

In  most  practical  problems  the  leakance  of  the  condenser  is 
negligible,  that  is,  </=0.  When  this  condition  holds,  the  con- 
stants in  equation  (39)  have  the  values 

r 
~2L 


I      \  ,     47,        I  1 

m= — \  r  —  —  =\  u2  — 

2Z/V          C     ^         LC 


R=r 


ALTERNATING  CURRENTS  357 


2.7T/C 

202.  Discharge  of  a  Condenser  having  Negligible  Leakance 
through  a  Resistance  and  Inductance.  —  As  an  example  of  the 
application  of  equation  (39),  T 

the  manner  in  which  a  con- 
denser of  capacity  C  (Fig.  108) 
charged  to  a  p.d.  V  discharges 
through  a  circuit  containing  a 
resistance  r  and  inductance  L 
will  be  determined.  (Compare 

i  f   ±1        j-  Fig.  108. 

with  the  ideal  case  of  the  dis- 
charge of  a  condenser  through  an  inductance  having  no  resistance, 
Article  164.)     At  time  t  =0,  the  switch  S  is  closed.     The  impressed 
e.  m.  f.  is  therefore  zero,  hence  the  equation  for  the  current  is 
i=tut  [Avtmt  +  A2tmi}  (a) 

where  Al  and  A2  are  constants  to  be  determined  from  the  initial 
current  and  p.d.  across  the  condenser. 

If  there  is  originally  no  current  in  the  circuit,  the  current  must 
start  from  zero  (otherwise  there  would  be  an  instantaneous  transfer 
of  a  finite  amount  of  energy,  iLi2,  to  the  magnetic  field  set  up  by 
the  current,  which  would  mean  an  infinite  rate  of  transfer,  or 
infinite  power,  which  is  impossible)  and  therefore  at  time  t=Q, 

{=0.     We  also  have  from  the  relation  v  +  ri+L — =0,  that  at 
time  t  =0, 

T  di     ran       v 

V  =  -L  —  or  =-- 

dt        L<ft;Jj.0     L 

Hence,  substituting  these  conditions  in  equation  (a),  we  have  that 

_7 
L 

Whence,  solving  these  two  equations,  we  get 

V  V 

and  Az  —  + 


2mL  2mL 

These  values  of  Al  and  A2  substituted  in  (a)  give 

V 


r-     ..-  ,  (40) 

2mL 


358 


ELECTRICAL  ENGINEERING 


Since  the  leakance  g  of  the  condenser  is  assumed  to  be  zero,  u 
and  m  have  the  values 

r 


u=- 


2L 


LC 
u  is  therefore  always  real,  but  m  may  be  either  real  or  imaginary, 

depending  upon  whether  u2  >  —  or  whether  uz  < ,  that  is,  upon 

LC  L/C 

2      4L        ,       4L 
whether  r2  >  —  or  r2  s  — . 

C  "  C 

For  r2  >  — ,  the  constant  m  is  real  and  equation  (40)  may 
C 

therefore  be  written 


mL 


"^  sinh  (mi) 


(40a) 


This  equation  tells  us  that  the  current  starts  at  zero,  rises  to 


Fig.  109. 


a  maximum  value  in  the  negative  direction  (corresponding  to 

—  =0)  and  then  decreases  to  zero.     The  shape  of  the  curve  repre- 
dt 

sented  by  equation  (40)  in  this  case  is  as  shown  in  Fig.  109. 


*The  symbol  sinh  x  is  used  to  represent  the  expression 
is  called  the  hyperbolic  sine  of  x. 


_  €- 


which 


ALTERNATING  CURRENTS 


359 


For    r2    <  — ,   we 

C 


have  m=  \l(— !)    ( —u2\ 

'  \LC        / 


or   m  = 


where  j 


—l  and  <H= 


—  u2,  a  real  quantity.     In  this  case 


equation  (40)  may  be  written 


=  — e~ut  sw 


(406) 


snce 


tu    _ 


2] 


=sm  (cat).     Equation   (406)   tells  us  that  when 


4L 


r2  <  :  _  the  current    in  the  circuit   oscillates  with  a  frequency 


Fig.  110. 


/=—  = — \ —  —  u2  but  that  the  amplitude  of  the  oscillations 

2?r     277^  LC 
decreases  as  time  increases,  as  shown  in  Fig.  110. 

4Z/ 
For  r2= — ,  the   constant   m=0  and  therefore  €mt  —  €-mt  [s 

u 
also    zero.     Hence    equation    (40)    for    the   current   is   indeter- 

mt e-m< 

minate.     Evaluating    the  indeterminate   expression we 


m 


get 


€       —  € 

m 


(f 

-  —  vfc 
dm 


mt 


dm 


ra=  0 


m-0 


m=  o 


60  ELECTRICAL  ENGINEERING 

lence  in  this  case  equation  (40)  becomes 

*'  =  — ^-  (40c) 

Li 

which  likewise  represents  a  non-oscillating  discharge  such  as  shown 

4  T 
in  Fig.  109.    When  r2  = —  the  current  just  ceases  to  be  oscillatory. 

Equation  (40c)  may  therefore  be  looked  upon  as  the  limiting  form 
of  the  oscillating  current  given  by  (406). 


SUMMARY    OF    IMPORTANT    DEFINITIONS 
AND   PRINCIPLES 

Note  :  The  definitions  given  in  paragraphs  1  to  9  inclusive  are 
in  terms  of  current;  they  also  apply  to  electromotive  forces  and 
potential  differences. 

1.  An   alternating   current   is    a   current   which    varies    con- 
tinuously with  time  from  a  constant  maximum  in  one  direction 
to  an  equal  maximum  in  the  opposite  direction  and  back  again 
to  same  maximum  in  the  first  direction,  repeating  this  cycle  of 
values  over  and  over  again  in  equal  intervals  of  time. 

2.  The  period  T  of  an  alternating  current  is  the  time  taken 
for  the  current  to  pass  through  a  complete  cycle  of  positive  and 
negative  values. 

3.  The  frequency  /  of  an  alternating  current  is  the  number 
of  complete  cycles  of  values  which  it  passes  through  in  one  second 


4.  The  number  of  alternations  a  per  minute  is  the  total  number 
of  times  per  minute  that  the  current  changes  in  direction 

f 

5.  The  equation  of  a  harmonic  current  of  maximum  value  /„  is 

i=I0  sin  (a)t  +  0) 

where  w,  called  the  periodicity  of  the  current,  is  equal  to  —  =2irf, 

and  6,  called  the  phase  of  the  current,  is  a  constant  such  that 
I0  sin  6  gives  the  value  of  the  current  at  time  t  =0. 

6.  The  difference  in  phase  between  a  harmonic  current  and  a 
harmonic  p.d.  is  the  angle  corresponding  to  the  time  between 


ALTERNATING  CURRENTS  361 

successive  maxima  values  of   the  current  and  p.d.  respectively. 

7.  The  equation  of  a  non-harmonic  current  of  frequency  / 
may  be  written 

where  cy=2?r/  and  the  Ps  and  0's  are  constants.  The  first  term 
in  this  expression  is  called  the  fundamental  or  first  harmonic, 
the  succeeding  terms  the  second,  third,  etc.,  harmonics. 

8.  The  instantaneous  value  of  an  alternating  current  is  its 
value   at   any   instant;  the  maximum  value   of   an  alternating 
current   is   its   greatest   instantaneous   value   during   any   cycle; 
the  average  value  of  an  alternating  current  is  the  numerical  value 
of  the  average  of  its  instantaneous  values  between  successive 
zero  values.     For  a  harmonic  current 

2 

7        —  —  I 

Aaver.         '     max. 
IT 

9.  The  effective  value  of  an  alternating  current  is  the  square 
root  of  the  mean  of  the  squares  of  its  instantaneous  values  over  a 
complete  period,  that  is,  j 


where  T  is  a  complete  period  of  the  current  and  i  its  instantaneous 
value  at  any  instant.  When  an  alternating  current  is  expressed 
as  so  many  amperes  this  effective  value  is  always  meant  unless 
specifically  stated  otherwise.  The  effective  value  of  a  harmonic 
current  is 

€ff     V2~ 

10.  When  a  harmonic  current  of  effective  value  I  and  a  har- 
monic potential  drop  of  the  same  frequency  and  of  effective  value 
V  exist  in  a  circuit,  the  average  electric  power  input  into  this 
circuit  during  each  cycle  is 

P  =  VI  cos  0 
where  0  is  the  difference  in  phase  between  the  current  and  p.d. 

11.  The  power  factor  of  a  circuit  is  the  ratio  of  the  average 
power  input  P  to  the  product  of  the  effective  value '  V  of  the  p.d. 
by  the  effective  value  I  of  the  current,  i.e., 

P 

Power  factor  =77; 


362  ELECTRICAL  ENGINEERING 

12.  The  average  electric  power  input  into  a  circuit  when  the 
current  and  p.d.  are  not  harmonic  functions  of  time,  or  sine  waves, 
is  equal  sum  of  the  values  of  the  average  power  corresponding 
to  each  pair  of  harmonics  of  the  same  frequency,  i.e., 

P  =  VJl  cos  6l  +  VJ2  cos  02  +  VJ3  cos  03  +  etc. 
where  the  V's  and  I's  are  the  effective  values  of  the  successive 
harmonics  of  the  p.d.'s  and  currents,  and  the  O's  the  difference 
in   phase   between   the   corresponding   harmonics    of    the   same 
frequency. 

13.  The  effective  value  of  a  non-harmonic  current  is  equal  to 
the  square  root  of  the  sum  of  the  squares  of  the  effective  values 
of  all  the  harmonics  present  in  the  current  wave,  i.e.,  is  equal  to 

7  =  V//12  +  /2-f/32+^c. 

where  /,,  72,  73,  etc.,  are  the  effective  values  of  the  harmonics. 
Similarly  the  effective  value  of  a  non-harmonic  p.d.  is 

V  =  Vy*  +  V*  + 


where  Vl}  V2,  V3,  etc.,  are  the  effective  values  of  the  harmonics 
in  the  p.d.  wave. 

14.  The  equivalent  sine-wave  p.d.  and  current  are  the  harmonic 
p.d.  and  the  harmonic  current  which  have  respectively  the  same 
effective  values  as  the  actual  p.d.  and  current  and  differ  in  phase 
by  the  angle  whose  cosine  is  equal  to  the  power  factor. 

15.  The  form  factor  of  a  wave  is  the  ratio  of  the  effective 
.value  to  the  average  value. 

16.  The  amplitude  factor  of  a  wave  is  the  ratio  of  the  maximum 
value  to  the  effective  value. 

17.  The  component  of  the  current  in  any  portion  of  a  circuit 
in  phase  with  the  potential  drop  through  this  portion  of  the 
circuit  is  called  the  power  component  of  the  current  in  this  portion 
of  the  circuit.     The  component  of  the  p.d.  in  phase  with  the 
current  is  called  the  power  component  of  the  p.d.     When  the 
current  and  p.d.  are  sine  waves  having  the  effective  values  V  and 
/  respectively  and  differing  in  phase  by  the  angle  6,  the  effective 
value  of  the  power  component  of  the  current  is  7  cos  6  and  the 
effective  value  of  the  power  component  of  the  p.d.  is    V  cos  6. 

The  component  of  the  current  in  quadrature  with  the  p.d. 
is  called  the  reactive  component  of  the  current  and  the  component 
of  the  p.d.  in  quadrature  with  the  current  is  called  the  reactive 
component  of  the  p.d.  For  sine-wave  current  and  p.d.  the 


ALTERNATING  CURRENTS  363 

effective  value  of  the  reactive  component  of  the  current  is  I  sin  0 
and  the  effective  value  of  the  reactive  component  of  the  p.d.  is 
V  sin  6. 

18.  Apparent  power,  volt-amperes  or  apparent  watts  is   the 
product  of  the  effective  value  of  the  p.d.  by  the  effective  value  of 
the  current. 

19.  Reactive  power  is  the  product  of  the  effective  value  of  the 
current  (or  p.d.)  by  the  effective  value  of  the  component  of  the 
p.d.  (or  current)  in  phase  with  it. 

20.  A  sine-wave  current  (p.d.  or  e.m.f.)  may  be  represented  by  a 
vector  equal  in  length  to  the  effective  value  of  the  current  (p.d.  or 
e.m.f.)  making  an  angle  with  an  arbitrarily  chosen  axis  of  reference 
equal  to  its  phase  angle. 

21.  The   average   power   corresponding   to   a  sine-wave   p.d. 
and  a  sine-wave  current  is  equal  to  the  product  of  the  length 
of  the  vector  representing  the  p.d.  by  the  projection  of  the  vector 
representing  the  current  on  the  p.d.  vector,  which  in  turn  is  equal 
to  the  product  of  the  length  of  the  current  vector  by  the  pro- 
jection of  the  p.d.  vector  upon  the  current  vector. 

22.  The  effective  value  of  the  potential  drop  due  to  a  sine- 
wave  current  of  effective  value  I  and  frequency  /  in  a  circuit 
having  a  constant  resistance  r  and  a  constant  inductance  L  is 


and  leads  the  current  by  the  angle 


23.  The  effective  resistance  r  of  any  portion  of  a  circuit  to  an 
alternating  current  is  the  ratio  of  the  average  rate  Ph  at  which 
heat  energy  is  developed  in  this  portion  of  the  circuit  to  the 
square  of  the  effective  value  I  of  the  current  in  this  portion  of 
the  circuit,  i.e., 


24.  The  impedance  z  of  any  portion  of  a  circuit  is  the  ratio  of 
the  numerical  value  of  the  vector  difference  of  the  externally 
induced  e.  m.  /.  E  and  the  resultant  p.d.  V  in  this  portion  of  the 
circuit,  both  in  the  direction  of  the  current,  to  the  effective  value 
/  of  the  current,  i.e., 


364  ELECTRICAL  ENGINEERING 


E-V 

z  =  —  — 

When  there  is  no  externally  induced  e.  m.  f. 

V 


z=  — 


25.  The  reactance  x  of  any  portion  of  a  circuit  is  the  square 
root  of  the  difference  between  the  squares  of  the  impedance 
and  the  reactance  of  the  given  portion  of  the  circuit,  i.e., 


x    =z-r 

The  reactance  is  taken  positive  when  the  current  lags  behind 
the  potential  drop  in  the  direction  of  the  current,  negative  when 
the  current  leads  this  p.d. 

26.  In  the  case  of  a  sine-wave  current  and  no   externally 
induced  e.  m.  /.  the  angle  by  which  the  current  lags  behind  the 
p.d.  is 

0=tanlX- 
r 

27.  The  impedance  of  a  circuit  of  constant  resistance  r  and 
inductance  L  to  a  sine-wave  current  of  frequency  /  is 


and  its  reactance  is 

X=27TfL 

28.  The  impedance  of  a  condenser  of  capacity  C  and  leakance 
g  to  a  sine-wave  current  of  frequency  /  is 

1 


z—- 


and  its  reactance  is 

27T/C 


29.  The  impedance  of  a  circuit  formed  by  a  resistance  r,  an 
inductance  L  and  a  capacity  C  in  series  to  a  sine-wave  current  is 


and  the  reactance  is 

X=2<7TfL ?-_ 

27T/C 

provided  the  condenser  of  capacity  C  has  no  leakance. 


ALTERNATING  CURRENTS  365 

30.  A  circuit  is  said  to  be  in  resonance  with  the  impressed 
electromotive  force  when  the  frequency  of  the  impressed  e.  m.  /. 
is  the  same  as  the  'natural  or  free  frequency  of  the  circuit.  The 
free  frequency  of  a  closed  circuit  formed  by  a  capacity  C  and  an 
inductance  L,  when  the  resistance  and  leakance  are  negligible,  is 

1 


31.  In  a  circuit  formed  of  two  or  more  impedances  in  series, 
the  resultant  effective  resistance  is  the  arithmetical  sum  of  the 
separate  effective  resistances,  the  resultant  reactance  is  the 
algebraic  sum  of  the  separate  reactances,  and  the  resultant 
impedance  is  the  square  root  of  the  sum  of  the  squares  of  the 
resultant  resistance  and  reactance,  provided  the  current  and 
p.d.  are  sine  waves  ;  that  is 


32.  The  effective  conductance  g  of  any  part  of  a  circuit  to  an 
alternating  current  is  the  ratio  of  the  average  rate  Ph  at  which 
heat  energy  is   developed  in  this  portion  of  the  circuit  to  the 
square  of  the  effective  value  V  of  the  p.d.  through  it,  i.e., 

9=^ 
V2 

33.  The  admittance  y  of  any  part  of  a  circuit  is  the  ratio  of  the 
effective  value  of  the  current  7  in  this  portion  of  the  circuit  to  the 
numerical  value  of  the  vector  difference  of  the  externally  induced 
e.  m.  f.  E  and  the  resultant  p.d.  V  in  this  portion  of  the  circuit, 
both  in  the  direction  of  the  current,  i.e., 

y==E^V 
When  there  is  no  externally  induced  e.  m.  /. 

yjv 

34.  The  susceptance  of  any  portion  of  a  circuit  is  square  root 
of  the  difference  between  the  squares  of  the  impedance  and  the 
reactance  of  this  portion  of  the  circuit,  i.e., 


The  susceptance  is  taken  positive  when  the  current  lags  behind 


366  ELECTRICAL  ENGINEERING 

the  potential  drop  in  the  direction  of  the  current,  negative  when 
the  current  leads  this  p.d. 

35.  In  the  case  of   a   harmonic   current  'and   no   externally 
induced  e.  m.  /.  the  angle  by  which  the  current  lags  behind  the  p.d. 
is 

0=tanl- 
9 

36.  The  resistance  r,  the  conductance  g,  the  reactance  x,  the 
susceptance  b,  the  impedance  z  and  the  admittance  y  of  a  given 
portion  of  a  circuit  when  the  current  and  p.d.  are  sine  waves 
are  related  as  follows: 


1  1 

*--  y=~ 

y  z 

r=g-  g=- 

y2  * 

b  X 


37.  The  admittance  of  a  condenser  of  capacity  C  and  leakance 
to  a  sine-wave  current  of  frequency  /  is 


and  its  susceptance  is 

b=-27TfC 

38.  In  a  circuit  formed  of  two  or  more  impedances  in  parallel, 
the  resultant  effective  conductance  is  the  arithmetical  sum  of 
the  separate  effective  conductances,  the  resultant  susceptance 
is  the  algebraic  sum  of  the  separate  susceptances,  and  the  resultant 
admittance  is  the  square  root  of  the  sum  of  the  squares  of  the 
resultant  effective  conductance  and  susceptance,  provided  the 
current  and  p.d.  are  sine  waves,  that  is 


39.  The  discharge  of  a  condenser  of  capacity  C  but  no  leak- 
ance through  a  resistance  r  and  an  inductance  L  in  series  is 
oscillatory  when 


ALTERNATING  CURRENTS  367 

,      4L 
and  non-oscillatory  when 


PROBLEMS 

1.  The  equation  of  an  alternating  current  is  t=100  sin  377t. 
Find  (1)  the  maximum  value  of  the  current,  (2)  the  rate  at  which 
the  current  is  changing  when  the  current  is  a  maximum,  (3)  the 
rate  at  which  the  current  is  changing  when  the  current  is  zero, 
(4)  the  effective  value,  (5)  the  average  value,  (6)  the  frequency, 
(7)  the  period,  and  (8)  the  periodicity. 

Ans.:  (1)  100  amperes;  (2)  0  amperes  per  second;  (3)  37,700 
amperes  per  second;  (4)  70.7  amperes;  (5)  63.7  amperes;  (6) 
60  cycles  per  second;  (7)  0.01667  second;  (8)  377  radians  per 
second. 

2.  Find  (1)  the  effective  value,  (2)  the  average  value,  (3)  the 
form  factor,    and    (4)  the    amplitude  factor  of  a    semi-circular 
shaped  wave,  the  maximum  value  of  which  is  A. 

Ans.:  (1)  0.816  A;  (2)  0.785  A;  (3)  1.04;  (4)  1.23. 

3.  If  a  direct  current  of  10  amperes  and  an  alternating  current 
of  10  amperes  (effective)  exist  at  the  same  time  in  a  circuit,  what 
will  an  alternating  current  ammeter  connected  in  series  with 
this  circuit  indicate? 

Ans.:  14.14  amperes.  v 

4.  The  equation  of  the  current  in  a  circuit  is 

i =50  sin  (CD  I  +  20°)  +  30  sin  (3  ait- 15°)  +  10  sin  (5a)t  +  30°) 
and  the  equation  of  the  potential  drop  through  it  is 
^=100  sin  (a>  £-10°)+40  sin  (3  co  £-30°) 

Determine  (1)  the  effective  value  of  the  current,  (2)  the  effective 
value  of  the  potential  difference  across  the  circuit,  (3)  the  average 
power  absorbed  by  the  circuit,  (4)  the  power  factor  of  the  circuit, 
and  (5)  the  equivalent  sine  waves  of  current  and  potential 
difference. 

Ans.:  (1)  41.8  amperes;  (2)  76.1  volts;  (3)  2744  watts;  (4) 
86.3%;  (5)  i=59.2  sin  cat  and  v  =  W7.S  sin  (ut—  30.3°). 

5.  The  equations   of   the  potential   drop   through  the  three 
parts  of  a  series  circuit  are  vv  =80  sin  (ojt  +  60°),  v2  =60  sin  a)  t  and 


368  ELECTRICAL  ENGINEERING 

v3=5Q  sin  ((tit  —  80°).  Find  (1)  the  equation  of  the  potential 
drop  through  the  entire  circuit,  and  (2)  the  effective  value  of  the 
total  potential  drop. 

Ans.:  (1)  v=110.6  sin  (w£  +  10.5°);  (2)  78.2  volts. 

6.*  An  e.  m.  f.  of  150  volts,  the  frequency  of  which  is  60  cycles, 
is  impressed  upon  a  series  circuit  consisting  of  a  resistance  of  5 
ohms,  an  impedance  coil  of  2  ohms  resistance  and  0.1  henry  induc- 
tance and  a  condenser  of  50  microfarads  capacity.  Find  (1)  the 
value  of  the  current  established  in  the  circuit  and  the  potential 
drops,  (2)  through  the  5  ohm  resistance,  (3)  through  the  impedf 
ance,  and  (4)  through  the  condenser.  Find  the  angle  by  which 
the  current  lags  behind  the  potential  drop  (5)  through  the  5  ohm 
resistance,  (6)  through  the  impedance  coil,  (7)  through  the  con- 
denser and  (8)  through  the  entire  circuit.  Draw  a  complete 
vector  diagram. 

Ans.:  (1)  8.88  amperes;  (2)  44.4  volts;  (3)  336  vplts;  (4) 
472  volts ;  (5)  0° ;  (6)  87.0° ;  (7)  -  90° ;  (8)  -  65.6°. 

7.  (1)  What  is  the  maximum  60-cycle  e.  m.  f.  which  may  be 
impressed  upon  a  circuit  formed  by  an  impedance  of  0.1   ohm 
resistance  and  0.3  henry  inductance  and  a  condenser  of  23.45 
microfarads  capacity  connected  in  series,  if  the  effective  punctur- 
ing voltage  of  the  condenser  dielectric  is  10,000  volts?    (2)  If  a 
25-cycle  e.  m.  f.  of  the  same  value  as  this  60-cycle  e.  m.  f.  determined 
above  is  impressed  upon  this  circuit,  what  is  the  potential  drop 
across  the  condenser? 

Ans.:  (1)  8.84  volts;  (2)  10.69  volts. 

8.  When  an  e.  m.  f.  of  220  volts  is  impressed  upon  a  circuit 
formed  by  two  impedances  A  and  B  connected  in  series,  the  power 
absorbed  by  A  is  300  watts  and  the  power  absorbed  by  B  is 
1200  watts.     If  the  potential  drop  across  A  is  200  volts  and  the 
current  in  A  is  10  amperes,  find  (1)  the  potential  drop  across 
B}  (2)  the  resistances,  and  (3)  the  reactances  of  A  and  B  re- 
spectively. 

Ans.:  (1)  125.4  volts;  (2)  378  volts;  (3)  3  and  12  ohms;  (4) 
±  19.8  ohms  for  A  and  ±  3.67  or  ±  35.9  ohms  for  B. 

9.  The  current  established  in  a  circuit  consisting  of  a  resist- 
ance of  3  ohms  in  series  with  an  impedance  of  unknown  resistance 

\      • 

*The  values  of  the  e.  m.  /.,  potential  difference,  and  current  in  Problem 
6  and  in  all  problems  following  are  effective  values  and  the  wave  forms 
sinusoidal. 


ALTERNATING  CURRENTS  369 

and  reactance  is  12  amperes.  If  the  power  absorbed  by  the 
circuit  is  600  watts  and  the  potential  drop  across  the  impedance 
is  100  volts,  find(l)  the  impressed  voltage  on  the  circuit,  (2)  the 
power  factor  of  the  circuit,  and  (3)  the  angle  by  which  the  drop 
through  the  impedance  leads  the  current. 
Ans.:  (1)  111  volts;  (2)  45.1%;  (3)  82.0°. 

10.  Two  impedances  A  and  B  are  connected  in  parallel.     A 
has  a  resistance  of  5  ohms  and  an  inductance  of  0.02  henry,  while 
B  is  formed  by  a  resistance  of  10  ohms  and  a  capacity  of  100 
microfarads  in  series.     If  a  25-cycle  e.m.f.  of  220  volts  is  impressed 
upon  this  parallel  circuit  find  (1)  the  total  current,  (2)  the  current 
in  each  branch,  and  (3)  the  power  factor  of  the  circuit. 

Ans.:  (1)  36.1  amperes;  (2)  37.3  and  3.41  amperes  respec- 
tively; (3)  89.2%. 

11.  At  60  cycles  the  impedances  of  two  coils  A  and  B  are 
each  10  ohms.     At  25  cycles  the  impedance  of  A  is  5.38  ohms 
and  the  impedance  of  B  is  8.67  ohms.     Find  the  joint  impedance 
of  A  and  B  at  60  cycles  when  connected  (1)  in  parallel,  and  (2)  in 
series. 

Ans.:  (1)  5.24  ohms;  (2)  19.06  ohms. 

12.  A  parallel  circuit  consists  of  a  coil  (A)  of  negligible  resist- 
ance and  0.05  henry  inductance  connected  in  parallel  with  a 
condenser  (B)  of  140.7  microfarads  capacity  and  negligible  leak- 
ance.     This  parallel  circuit  is  connected  in  series  with  an  im- 
pedance (C)   of  2  ohms  resistance  and  0.02  henry  inductance. 
Find  (1)   the  current,  and  (2)   the  potential  drop  in  A,  B  and 
C  respectively,  when  an  e.  m.  /.  of  220  volts  and  60  cycles  is  im- 
pressed upon  this  circuit.     If  a  non-inductive  resistance  ( D)  of 
10  ohms  is  connected  in  parallel  with  (A),  find  (3)  the  current  and 
(4)  the  potential  drop  in  A,  B,  C  and  D  respectively.     Draw  a 
vector  diagram  for  each  case. 

Ans.:  (1)  11.67,  11.67  and  0  amperes ;  (2)  220,  220  and  0  volts  ; 
(3)  8.24,  8.24;  15.53  and  15.53  amperes; (4)  155.3,  155.3,  121.0  and 
155.3  volts. 

13.  When  a  50-kw.,  60-cycle,  a.c.  generator  is  delivering  30  kw. 
at  80%  power  factor  the  terminal  voltage  is  230  volts.     The 
effective  resistance  of  the  armature  between  terminals  is   0.05 
ohm  and  the  reactance  0.1  ohm.     Neglecting  the  effect  of  armature 
reaction,  (1)  what  would  be  the  terminal  voltage  of  this  machine 
if  the  external  circuit  were  opened?    (2)  What  is  the  maximum 


370  ELECTRICAL  ENGINEERING 

load  under  which  this  generator  could  operate  continuously  at 
80%  power  factor? 

Ans.:  (1)  246.5  volts;  (2)  40  kw. 

14.  500  kw.  are  delivered  to  a  substation  from  a  6600- volt 
power  station  over  a  transmission  line  of  6  ohms  resistance.     If 
the  current  taken  by  the  load  is  100  amperes,  find  (1)  the  power 
factor  at  the  power  station,  and  (2)  the  efficiency  of  the  transmis- 
sion line. 

Ans.:  (1)  84.9%;  (2)  89.3%. 

15.  The  potential  differences  at  the  load  and  generator  ends  of 
a  transmission  line  are  each  6600  volts.     The  resistance  of  the 
line  is  6  ohms  and  the  reactance  8  ohms.     If  the  line  current  is 
100  amperes,  find  (1)  the  power  factor  at  the  load,  and  (2)  the 
efficiency  of  the  transmission  line. 

Ans.:  (1)75.2%;  (2)  89.2%. 

16.  A  transmission  line  25  miles  in  length  consists  of  two 
No.  0  B.  &  S.  wires  (diameter  0.325  inch)  spaced  3  feet  apart. 
At  the  end  of  this  line  is  connected  a  60-cycle  induction  motor 
load  (lagging  current)  of  1500  kw.  operating  at  25,000  volts  and 
at  a  power  factor  of  75%.     What  must  be  the  potential  difference 
at  the  power  station  supplying  this  load?    The  leakage  and  ca- 
pacity of  the  line  are  to  be  neglected. 

Ans.:  28,400  volts. 

17.  Two  220  alternators  A  and  B  connected  in  parallel  supply 
an  inductive  load  of  50  kw.  at  90%  power  factor.     If  A  supplies 
one-third  of  this  load  at  a  power  factor  of  80%,  find  (1)  the  power 
factor  at  which  B  supplies  power  to  the  load,  and  (2)  the  armature 
current  of  each  alternator,  if  both  currents  lag  behind  the  gen- 
erated e.  m.  /.'s. 

Ans.:  (1)  94.3%  ;  (2)  94.7  and  160  amperes. 


VIII 

SYMBOLIC   METHOD  OF   TREATING   ALTERNATING 

CURRENTS 

203.  Symbolic  Representation  of  a  Vector. — Let  A  =  0  P  in  Fig. 
Ill  be  any  vector  making  an  angle  6  with  an  arbitrary  line  of 
reference  0  X.  The  component  of  A  in  the  direction  of  0  X  is 


\ 


A2  =  A  sin  6    \ 
X 


A!=  A  cos  0 
Fig.  ill. 

then  A  cos  0  and  the  component  of  A  90°  ahead  of  this  line  of 
reference  is  A  sin  8.  Put 

A  cos  6  =  A,  (la) 

A  sin  0  =  A2 

Employing  the  usual  convention  of  indicating  a  vector  sum  we 
may  write  A=Al+A2}  where  the  line  over  Av  and  A2  indicates 
that  Al  and  A2  are  to  be  added  vectorially.  Instead  of  using 
this  symbol,  however,  we  may  in  the  special  case  of  two  mutually 
perpendicular  vectors  represent  the  vector  A  by  the  symbolic 
expression 

A=Al+jA2  (16) 

where  the  symbol  /  indicates  that  the  vector  A2  leads  the  line  of 
reference  by  90°,  while  Al  coincides  in  direction  with  the  line 
of  reference. 

When  a  single  letter  is  used  to  represent  a  vector  in  this 
symbolic  notation,  it  is  usual  to  write  a  dot  under  the  letter. 
That  is,  the  letter  A  by  itself  represents  the  length  of  the  vector, 
while  A  represents  the  vector  expressed  in  this  symbolic  notation. 
Squaring  and  adding-  equations  (la)  and  taking  the  square 
root,  we  get 

371 


372 


ELECTRICAL  ENGINEERING 


Therefore,  the  length  of  a  vector  is  equal  to  the  square  root  of 
the  sum  of  the  squares  of  the  two  components  entering  into  its 
symbolic  expression.  Taking  the  ratio  of  two  equations  (la)  we  get 

tanO=—  (Id) 

Whence,  the  angle  which  a  vector  makes  with  the  line  of  reference 
is  equal  to  the  angle  whose  tangent  is  the  ratio  of  the  second  (or  j 
component)  to  the  first  component  in  its  symbolic  expression. 

Similarly,  any  other  vector  A'  =  OP'  making  an  angle  6' 
with  the  same  line  of  reference  may  be  resolved  into  the  two 
components  AJ  =A'  cos  9f  and  A2' =  A'  sin  0',  and  we  may  write 


where  j  as  before  indicates  that  the  vector  Azf  leads  the  line  of 
reference  by  90°  while  A'  coincides  in  direction  with  the  line  of 
reference. 

204.  Addition  of  Vectors.  — From  Fig.  112  it  follows  that  the 
resultant  of  two  vectors  A  and  A'  is  the  vector  which  has  the  com- 


.»         •" 


Fig.  112. 

ponent  A  cos0+A'  cos0f=Al  +  A1/  parallel  to  the  axis  of  reference 
and  the  component  A  sin  9+  A'  sin  9'  =A2+  A2'  leading  the  line 
of  reference  by  90°.  The  resultant  of  the  two  vectors  A  and  A' 
is  then  represented  symbolically  as 

S  =  A  +  A'=(A1  +  A1')+j(A2+A2')  (2a) 

The  length  of  this  vector  is  then 

8  =  V(A,  +  A 
and  it  makes  an  angle  .  , 

~l 


(26) 


SYMBOLIC    METHOD 


373 


with  the  axis  of  reference.     Similarly,  the  sum  of  any  number  of 
vectors  is,  in  the  symbolic  notation, 


etc.) 
The  length  of  this  resultant  vector  is 


and  the  angle  which  it  makes  with  the  axis  of  reference  is 


s 


206.    Subtraction  of  Vectors.  —  The  vector  A  —  A'  (Fig.  113) 
similarly  the  vector  which  has  the  component  A  cos  0—  A'  cos  9' 


-A', 


\ 


-A'. 


V'-'' 


Fig.  1 13. 


=  Ai—  AS  in  the  direction  of  the  line  of  reference  and  the  com- 
ponent A  sin  0—  A'  sin  6'  =  A2—  A2f  leading  the  line  of  reference 
by  90°.  That  is,  the  vector  difference  A-  A'  is  represented  sym- 
bolically as 

D=A-A'=(Al-Al')+j(A2-A2')  (3a) 

The  length  of  this  vector  is  then 

D=V(A,-Atf  +  (A*-Atf  (36) 

And  it  makes  the  angle 


with  the  axis  of  reference.     When  this  angle  comes  out  negative, 
the  resultant  vector  makes  a  negative  angle  with  the  axis  of 


374 


ELECTRICAL  ENGINEERING 


reference  or  lags  behind  this  axis;  Fig.  113  illustrates  such  a  case. 
206.   The  Symbol  "7  "  as  a  Multiplier  Signifying  Rotation.  — 

So  far,  we  have  considered  the  symbol  "  j  "  simply  as  a  means  of 
indicating  the  component  of  a  vector  leading  the  line  of  reference. 
As  we  shall  presently  see,  it  is  also  frequently  convenient  to  look 
upon  "  7  "  as  a  multiplier  as  well  as  a  symbol  representing  direc- 
tion, and  to  define  the  operation  jxA}  where  A  is  the  symbolic 
expression  for  any  vector,  as  equivalent  to  turning  the  vector  A 
through  an  angle  of  90°  in  the  positive  direction.  This  convention 
leads  to  a  useful  mathematical  equivalent  for  the  symbol  "  /." 

Let  A  (Fig.  114)  be  a  vector  coinciding 
in  direction  with  the  line  of  reference. 
Then  from  the   above  convention    jA 
represents  a  vector  of  length  A  leading 
»     £         the  line  of  reference  by  90°.     Multiply- 
Fi«. 1 14-  ing    the    symbolic    expression    jA    for 

the  second  vector  by  7,  we  have  from  the  above  convention  that 
7X7*4  represents  a  vector  of  length  A  making  an  angle  of  90° 
with  the  vector  jA,  that  is,  a  vector  equal  in  length  to  A,  but 
in  the  opposite  direction  to  A.  Therefore 

jX]A=-A 

or  f  =  —  1 

whence 

J=V~1.  (4) 

It  should  be  borne  in  mind,  however,  that  although  the  sym- 
bol 7  is  mathematically  equal  to  the  imaginary  quantity  V— 1, 
the  physical  meaning  of  7  is  not  imaginary  at  all;  it  is  simply 
an  abbreviation  placed  before  the  expression  for  a  vector  signify- 
ing that  the  vector  in  question  is  to  be  turned  through  90°  in  the 
positive  direction.;  turning  a  vector  through  90°  twice  in  the 
same  direction  is  mathemati- 
cally equivalent  to  writing  a 
negative  sign  before  the  vec- 
tor. For  example,  in  the 
expression 


-A2  A, 

for  a  single  vector,    At  and  Fie-  115- 

A.,    both    represent    vectors    coinciding    in    direction  with    the 
line  of  reference,  but  in  the  expression  for  the  resultant  vec- 


SYMBOLIC    METHOD  375 

tor  A;  ^e  component  A2  is  turned  through  +  90°.  Again, 
when  the  resultant  vector  A=A1  +  jA2  is  turned  through  90°, 
the  expression  for  this  vector  in  its  new  position  is  jAl  —  Aa, 
as  is  readily  seen  from  Fig.  115.  But 


^ 
\ 

' 


since  f  =  —  1. 

207.  Symbolic  Expression  for  a  Vector  Referred  to  Any  Other 
Vector  as  the  Line  of  Reference.  —  Let  the  vector  A'  (Fig.  116) 
lead  the  vector  A  by  0  degrees;  it 
is  desired  to  write  the  symbolic  ex- 
pression  for  A'  referred  to  the  vector 
A  as  the  line  of  reference.     The  com-  A 

ponent  of  A'  parallel  to  A  is  A'  cos  9  Fig>  116' 

and  the  component  of  A'  leading  A  by  90°  is  A'  sin  9.     Hence,  the 
symbolic  expression  for  A'  referred  to  A  as  the  line  of  reference  is 
A'  =  A'  (cos  9  +  j  sin  9}  (5a) 

Similarly,  the  symbolic  expression  for  the  vector  A  referred 
to  Af  as  the  line  of  reference  is 

A=A(CosO-jsinO)  (56) 

since  A  lags  behind  A'  by  the  angle  9. 

For  example,  let  the  vector  A'  have  the  length  10  and  let 
this  vector  lead  a  second  vector  A  by  30°.  Then  the  symbolic 
expression  of  A'  referred  to  A  as  the  axis  of  reference  is 

4'  =10  (cos  30°  +  /  sin  30°) 
' 


If  the  vector  A'  lags  behind  the  vector  A  by  30°,  then  its  symbolic 
expression  referred  to  A  as  the  axis  of  reference  is 
A'  =10  (cos  30°  -/  sin  30°) 

=8.65-/  5 

208.  Difference  in  Phase  Between  Two  Vectors  Expressed  in 
Symbolic  Notation.  —  When  the  symbolic  expression  for  a  vector  A 
referred  to  any  given  line  of  reference  is 


the  angle  by  which  this  line  leads  the  line  of  reference  is 

9=tan^ 
At 

Similarly,  when  the  symbolic  expression  for  any  other  vector  A'  is 

A'=A/  +  /A2' 
the  angle  by  which  this  vector  leads  the  line  of  reference  is 


376 


ELECTRICAL  ENGINEERING 


Al 

Hence  the  angle  by  which  A'  leads  A  is 


-tan1—2 
A1 


or 


tan(9'-9}=lA2~^Az 

A     \   /    i      \     A   r 


tan  (('-      = 


Therefore  A'  leads  A  by 


8X5+6X4 

f  -  =  38°  40' 
5 


(6c) 


For  example,  let  A=8  +  j6  and  A' =  5  +  ?' 4.      Then  A'  leads 
A  by  the  angle  9'  —  9  where 


209.    Symbolic   Representation   of   a    Harmonic   Function.  - 

We  have  seen  (Articles  182  and  186)  that  a  harmonic  alternating 
current  i=^2  I  sin  (cot  +9),  where  /  is  the  effective  value  of  the 
current,  may  be  written 

i=V2  I  cos  9  sin  cot  +  ^2  1  sin  9  cos  cot 


and  that  the  terms 


cos  9  sin  cot  maybe  represented  by  a  vector 
equal  in  length  to  7t=7  cos  9 
rotating  about  a  fixed  point  0 
with  the  angular  velocity  co 
and  the  term  V  2  /  sin  0  cos  cot 
by  a  second  vector  of  length 
72  =7  sin  0,  leading  the  first  vec- 
tor by  90°  and  rotating  about 
'this  same  point  with  the  same 
angular  velocity.  Hence  the 


Fi«-  117« 
function  i=\/2  1  sin  (cot  +  0)  may  be  represented  by  a  vector 


(7) 

rotating  about  this  point  with  the  angular  velocity  co,  where  /, 
is  a  vector  of  length  I  cos  0}  and  jI2  a  vector  of  length  I  sin  9 
leading  I,  by  90°,  both  rotating  about  the  same  point  with  the 
same  angular  velocity  co.  Therefore  we  may  look  upon  equation 
(7)  as  the  symbolic  expression  for  the  function  i  =  V  2  I  sin  (cot  +  9}. 
Note,  however,  that  the  two  vectors  7t  and  jI2  though  constant 
in  length  are  continually  changing  in  direction;  therefore  in  the 
expression 


SYMBOLIC  METHOD  377 


both  7j  and  I2  and  therefore  I  also,  must  be  looked  upon  as  functions 
of  time  and  not  as  constants,  since  a  vector  by  definition  is  a  quantity 
having  direction  as  well  as  magnitude. 

Note  the  analogy  with  a  body  moving  in  a  circle  of  radius  a 
with  a  constant  linear  speed  s;  the  velocity  of  such  a  body  is 
constant  in  magnitude,  but  is  continually  changing  in  direction. 

s2 

Hence  the  rate  of  change  of  the  velocity  is  not  zero,  but  is  —  ,  and 

r 

the  direction  of  this  rate  of  change,  i.e.,  of  the  acceleration,  is 
perpendicular  to  the  direction  of  the  velocity. 

210.  Symbolic  Expression  for  the  Derivative  of  a  Harmonic 
Function.  —  Since  a  harmonic  function  of  the  form 


may  be  written  in  the  form 

i  =  V  2  I  cos  9  sin  vt  +  Vz  I  sin  0  cos  cot 
the  derivative  of  i  with  respect  to  time  is 

di  /  — 

—  =  V  2  oj  I  cos  0  cos  w  £  —  v  2  Q)  /  sin  u  sin  cot 

at 

But  \/2  col  cos  6  cos  o)t  is  represented  by  a  vector  equal  in  length 
to  w  I  cos  0=0)1!  leading  by  90°  the  vector  7t  which  represents 
\/2  /  cos  6  sin  a)t;  and  —  \/2  col  sin  0  sin  ut  is  represented  by  a 
vector  of  length  CD  I  sin  0  =  co  I2  leading  by  90°  the  vector  y/2 
which  represents  i/2  I  sin  0  cos  cot.  Hence  the  derivative  of 
the  function  represented  by  the  vector 


di 

-•=/  oj  (/!+//,)=/  01  7  (8) 

at 

That  is,  the  derivative  with  respect  to  time  of  a  vector  rotating 
with  a  velocity  a>  results  in  a  vector  equal  in  length  to  the  product 
of  a)  by  the  length  of  the  original  vector,  which  vector  has  a 
direction  90°  ahead  of  the  original  vector. 

Vice  versa,  from  equation  (8)  we  also  have  that 

,7.1  f  (8a) 

•     CD  dt 

that  is,  multiplying  a  rotating  vector  by  j  is  equivalent  to  differ- 
entiating the  vector  with  respect  to  time  and  dividing  by  co. 


378  ELECTRICAL  ENGINEERING 

From  this  signification  of  the  multiplier  /  it  is  at  once  evident 
that  to  consider  the  multiplication  of  the  expressions  representing 
two  rotating  vectors  at  right  angles  to  each  other,  such  as  7  and 
jV  as  equivalent  to  /  (IV)  is'inconsistent  with  /  as  being  equiva- 
lent to  the  operation  -  -^.  For  —  (77)  =7  —  +  V  — ,  and  there- 
o)  dt  dt  dt  dt 

fore  to  be  consistent  with  equation  (8a),  the  expression  j  (IV) 
must  equal  7/7+  7/7. 

Consequently,  although  the  product  of  the  two  expressions 
I\  +  /^2  and7j  +  /  72  representing  the  current  and  voltage  in  a  circuit 
is  IiVi  —  7272  +  /  (7x72  +  7.27!),  when  /  is  considered  simply  as  a 
multiplier  equal  to  v  —  1,  this  expression,  however,  is  inconsistent 
with  the  meaning  of  /,  as  defined  by  equation  (8a),  since  Ilf  I2,Vl 
and  72  are  all  rotating  vectors.  It  is  important  to  bear  this 
clearly  in  mind,  since  one  is  likely  to  make  the  mistake  of  taking 
this  product  as  the  symbolic  expression  for  the  power  corre- 
sponding to  the  current  71+/72  and  the  p.d.  7i  +  /72,  and  to  take 
the  "real"  part  of  this  product  as  representing  the  average  power. 
As  a  matter  of  fact,  the  value  of  the  average  power  correspond- 
ing to  the  current  Ii+jI2  and  the  p.d.  7,  +  /72  is 

(see  Article  215).  That  is,  the  sign  between  the  two  terms  in  the 
expression  for  the  average  power  is  just  the  opposite  of  the  sign 
resulting  from  the  multiplication  of  7x  +  /72  and  71  +  /72  and  tak- 
ing /  as  a  multiplier  equal  to  ^  —  i. 

211.    Symbolic   Notation  for   Impedance. — Impedance   as   a 
Complex  Number. —  When  the  current  and  p.d.  in  a  circuit  of 
constant  resistance  r  and  constant  inductance  L  are  both  harmonic 
functions  of  the  same  frequency  they  may  be  represented  respective- 
ly by  the  two  vectors  7  and  V. 
The  component  of  the  result- 
.    ant  p.d.  in  phase  with  the  cur- 
rent is  a  vector  coinciding  in 
direction  with  7  and  equal  in 
i       length  to  the  product  of  r  by 
Fig.  us.  the  length  of  the  current  vec- 

tor, that  is,  is  represented  by 

the  vector  rl  in  Fig.  118.  The  component  of  the  resultant  p.d.  90° 
ahead  of  the  vector  7  is  a  vector  leading  the  current  vector  by 


SYMBOLIC  METHOD  379 

90°  and  equal  in  length  to  the  product  of  the  reactance  x  by  the 
length  of  the  current  vector,  that  is,  is  represented  by  the  vector 
jxl.  Hence  the  resultant  p.d.  is  represented  by  the  vector 

V=rl  +  jxl  =  (r  +  jx)  I  (9) 

This  relation  also  follows  directly  from  the  differential  equation 

V=rI+L_,=rI+Ljo)I=rI  +  jxI  =  (r  +  jx)  I 

dt      ' 

for,  since  L  is  a  constant,  Lj  a)  =  j  aj  L  =jx. 

When  the  current  vector  is  referred  to  any  arbitrary  line  of 
reference  rotating  with  the  same  angular  velocity  as  the  current 
vector,  the  symbolic  expression  for  the  current  vector  is  7=/1  +  /72 
where  A  is  the  component  of  7  parallel  to  this  line  of  refer- 
ence and  72  the  component  of  7  leading  this  line  of  reference  by 
90°.  The  component  1^  flowing  through  the  resistance  r  produces 
a  drop  of  potential  rlj,  in  the  phase  with  7;  this  component  7j  flow- 
ing through  the  reactance  x  also  produces  a  drop  of  potential 
#7,  leading  7X  by  90°.  The  component  72  flowing  through  the 
resistance  r  produces  a  drop  of  potential  r72  in  phase  with  72  and 
therefore  leading  7t  by  90°,  since  72  leads  7:  by  90°;  this  com- 
ponent 72  flowing  through  the  reactance  x  also  produces  a  drop 
of  potential  xI2  leading  72  by  90°  and  therefore  leading  7X  by  180°. 
Hence  the  total  drop  of  potential  in  phase  with  II  is  ( 
and  the  total  drop  of  potential  90°  ahead  of  7t  is  (x 
Therefore,  in  symbolic  notation  the  vector  representing  the 
total  potential  drop,  referred  to  7X  as  the  line  of  reference,  is 


But  this  expression  is  exactly  the  same  as  would  result  from 
multiplying  (7t  +  /72)  by  (r+jx)  and  considering  /  as  equivalent 
to  V—  1,  that  is, 

V  =  (r+jx)  (/!  +  //,)  (9a) 

This  relation  also  agrees  with  the  interpretation  of  j  as  equiva- 
lent to  a  differentiation  with  respect  to  <u£,  for  since  r  and  x  are 
constants,  it  is  immaterial  whether  we  write  the  differentiation 
sign  before  or  after  these  constants.  Compare  with  the  significa- 
tion of  the  operation  represented  by  the  formula  (  Vl  +  JV2)  (I,  +  /72) 
where  both  terms  in  each  expression  are  rotating  vectors,  Article 
210. 

The  expression  r+jx  is  not  a  vector,  although  it  has  the  same 


380  ELECTRICAL  ENGINEERING 

mathematical  form  as  the  symbolic  expression  for  a  vector. 
Its  properties  in  any  operation  of  multiplication  or  division  are 

exactly  the  same  as  the  prop- 
erties of  a  " complex  number/' 
that  is,  a  number  which  con- 
j  *  sists  of  the  sum  of  two  terms, 
one  real  and  the  other  imagi- 
nary. Since  the  product  of  the 
symbolic  expression  for  cur- 
rent by  this  expression  r+jx 

gives  the  symbolic  expression  for  the  p.d.  due  to  this  current  in 
the  impedance  having  a  resistance  r  and  reactance  x,  the  expres- 
sion r-\-jx  may  be  looked  upon  as  the  symbolic  expression  for 
impedance,  and  may  be  represented  by  the  single  symbol  z  with 
a  dot  under  it,  i.e., 

z=r  +  jx  (10) 

We  may  then  write  the  .symbolic  expression  for  the  p.d.  produced 
by  a  current  7=/1  +  /72  in  an  impedance  z=r  +  jx  as 

V=zl  (11) 

Note  that  the  length  of  the  vector  represented  by  zl 
is  equal  to  Iz  and  that  this  vector  leads  the  vector  7  by  the 

angle  0=tari1  — .      Hence  multiplying  the  vector  7  -by  the   im- 
r 

pedance  z  gives  rise  to  a  vector  equal  in  length  to  the 
product  of  the  length  of  the  vector  7  by  the  numerical  value  of 
the  impedance,  which  vector  leads  7  by  the  power-factor  angle 
of  the  impedance. 

212.  Symbolic  Notation  for  Admittance.  —  The  relation  be- 
tween current  and  p.d.  when  both  are  harmonic  functions  of  the  same 
frequency  may  also  be  expressed  in  terms  of  the  admittance  of  the 
circuit.  Let  g  be  the  conductance  and  b  the  susceptance,  and  let 
7  and  V  be  the  current  and  p.d.  vectors.  Then  the  component  of 
7  in  phase  with  V  is  a  vector  coinciding  in  direction  with  the 
vector  V  and  equal  in  length  to  the  product  of  the  length  of  the 
p.d.  vector  by  the  conductance,  that  is,  is  represented  by  the 
vector  of  g  V  in  Fig.  120.  The  component  of  the  current  lagging 
behind  the  p.d.  by  90°  is  a  vector  lagging  behind  the  vector 
V  by  90°  and  equal  in  length  to  the  product  of  the  length 


SYMBOLIC    METHOD 


381 


of   the   p.d.  vector   by  the   susceptance,  that  is,  is  represented 
by  the  vector  -jbV;  since  +/  represents  a  lead  of  90°  and  there- 


fore   —  /  represents  a  lag  of  90°.     Hence  the  resultant  current 
vector  is 

In  the  particular  case  of  a  leaky  condenser  this  relation  also 
follows  from  the  differential  equation 


dt 

for  since  C  is  a  constant  Cjat V  =ja)CV  =  —  jbV.  In  the  case  of  a 
condenser  the  charging  current  coCV  leads  the  p.d.  by  90°,  and 
therefore  the  susceptance  of  the  condenser  is  —  CD  C,  since  the 
susceptance  of  the  circuit  is  the  factor  by  which  the  p.d.  must 
be  multiplied  to  give  the  component  of  the  current  lagging  90° 
behind  the  p.d. 

When  the  p.d.  is  expressed  as  Vl  +  jV3  the  operation  repre- 
sented by  (g— jb)(Vi  +  jV2)  can  also  be  shown  by  a  process  of 
reasoning  similar  to  that  employed  in  Article  211  to  be  equiva- 
lent to  the  product  of  g  —  jb  and  Vl  +  jV2  when  /  is  con- 
sidered as  equivalent  to  V—  1.  Therefore  in  symbolic  notation 
the  admittance  of  a  circuit  may  be  written  as  a  complex 
number 

y=g-fl>  (13) 

and  the  current  due  to  a  p.d.  V  across  this  admittance*  may  be 
written 

I=yV  (14) 

213.   Division  of  a  Rotating  Vector  by  a  Complex  Number. 

—  Since  by  equations  (11)  and  (14)  we  have 

V=zI  =  (r  +  jx)I  (a) 

l=yv=(g-fl>)  V  (b) 


382  ELECTRICAL  ENGINEERING 

the  division  of  the  vector   V  by  (r  +  jx)  must  be  equivalent  to 
multiplying  the  vector  V  by  (g—jb),  that  is 

V  I 

^=yV  or  -=y          (15) 

z      •  •  z      • 

Similarly,  the  division  of  the  vector  7  by  (g  —  jb)  must  be  equivalent 
to  multiplying  7  by  (r  +  jx)  ,  that  is 

•-=zl  or  -1=2          (16) 

y     '  '  y 

But  multiplying  equation  (a)  by  (r  —  jx),  considering  /  simply  as  a 

multiplier  equal  to  ^  —  1,  we  get 

(r-jx)  F=(r-2  +  x2)  / 

whence,  dividing  this  equation  by  r2  +  x2,  we  get 


which  to  be  consistent  with  1=  (g  —  jb)  V  requires  that 

r  x 

g=  and  6  = 


which  agree  with  the  relations  already  established  in  Article  199. 
Similarly,  multiplying  equation  (b)  by  (g  +  jb)  we  get 

(ff  +  fl>)  I  =  (g*  +  b2)  V 
whence,  dividing  this  equation  by 


which  to  be  consistent  with  V—(r  +  jx)  I  requires  that 

g 


r= 

g2+b2 

which  agree  with  the  relations  already  established  in  Article  199. 
In  general,  the  division  of  any  rotating  vector  A  by  a  com- 
plex number  of  the  form  a=a1±ja2  is  equivalent  to  multiplying 

the  rotating  vector  by  —  —  —,  that  is 


Consequently,  impedance  and  admittance  in  the  symbolic  notation 
may  be  treated  as  algebraic  multipliers  or  divisors,  the  symbol 
j  in  each  case  being  considered  as  mathematically  equivalent  to 


SYMBOLIC    METHOD  383 

v—l,  and  whenever  a  "  /  "  term  occurs  in  the  denominator  of  a 
fraction,  the  fraction  may,  by  the  relation  given  by  equation  (17), 
be  put  in  a  form  having  a  "  /  "  term  only  in  the  numerator. 

V 

Hence,  an  expression  of  the  form  —  =  —  ,  where  V  is  any  rotating 

tt1ia2  '  *  : 

vector  and  ai±/a2  is  any  complex  number,  represents  a  vector 

a,V 

having  the  component  -  =  —  in   phase  with   V  and  the  com- 

a2F  a'2  +  a*2 

ponent  q=  -  -  —  leading  V  by  90  . 


From  equations  (15)  and  (16)  the  equivalent  impedance  Z  of 
two  impedances  zl  and  z2  in  parallel  when  there  is  no  externally 
induced  e.  m.  f.  in  either  is 


For,   the  admittances   corresponding  to  zt  and  z2  are  yl=—  and 

zi 

y2  =—  ;  whence  the  equivalent  admittance  is  Y—y1  +  y2  =    +  —  = 

Z2  •  :      Zl       Z2 

•^  —  •-;  but  Z  —_  =_J_Ii..  In  any  numerical  example  this  expression 

Z,Z2  Y       ZL  +  ZZ 

is  readily  "rationalized"  by  applying  equation  (17).  Compare 
with  the  formula  in  Article  98  for  two  resistances  in  parallel 
when  there  is  no  electromotive  force  in  either  branch. 

214.  Kirchhoff's  Laws  in  Symbolic  Notation.  —  Kirchhoff's 
two  laws  for  the  relations  between  the  instantaneous  values  of 
the  current  and  electromotive  force  in  any  circuit,  may  then  be 
expressed  in  symbolic  notation  as  follows  : 

1.  The  sum  of  all  the  currents  flowing  to  any  point  in  any 
network  of  conductors  is  zero.     That  is,  at  any  point 

/  +  /'  +  /"  -----  =0  (18a) 

where  the  currents  are  all  expressed  in  symbolic  notation  and  are 
all  referred  to  the  same  line  of  reference. 

2.  The  sum  of  all  the  impedance  drops  in  a  given  direction 
around  any  closed  loop  in  any  network  of  conductors  is  equal  to 
the  sum  of  all  the  externally  induced  e.m.f.'s  (see  p.  339)  acting 
in  this  loop  in  this  direction.     That  is,  around  any  closed  loop 

(186) 


£84 


ELECTRICAL  ENGINEERING 


where  the  currents,  impedances,  and  electromotive  forces  are  all 
expressed  in  symbolic  notation,  and  the  currents  and  e.m.f.'s 
are  all  referred  to  the  same  axis  of  reference.  That  is,  the  currents 
'are  to  be  expressed  as  7=/1+//2>  /'  =//+///,  etc.  and  the 
e.m.f.'s  as  E  =  El  +  jI2,  Ef  =  El'  +j  E2'  ',  etc.  where  all  the  compo- 
nents of  the  currents  and  e.m.f.'s  with  the  subscript  1  are  parallel 
to  one  another  and  all  those  with  the  subscript  2  are  parallel 
to  one  another  and  lead  the  first  set  of  components  by  90°. 

The  electromotive  forces  due  to  inductance  and  capacity  are 
taken  account  of  by  the  impedance;  the  electromotive  forces 
represented  by  the  E's  in  equation  (186)  are  the  externally 
induced  electromotive  forces  such  as  those  due  generators  or 
motors  or  to  the  mutual  inductance  of  the  two  windings  of  a 
transformer. 

In  applying  equations  (18)  to  the  calculation  of  the  currents 
and  p.d.'s  in  any  network  of  circuits,  care  must  be  taken  to  desig- 
nate clearly  the  sense  of  the  vectors  representing  the  currents  and 

e.  m.  f.'s.  This  is  most  con- 
veniently done  by  number- 
ing all  the  junction  points 
in  the  network,  and  desig- 
nating each  current  and  e. 
m.  f.  by  a  double  subscript 
written  in  the  order  corre- 
sponding to  the  assumed 
direction  of  the  current  or 
e.  m.  /.  vector.  For  exam- 
ple, in  Fig.  121,  the  e.m.f. 
from  0  to  1  is  represented  by 
Eol  while  the  e.  m.  /.  from  1 
to  0,  which  is  equal  to-EQl  is  represented  by  Elo.  In  the  figure, 
then,  the  net  electromotive  force  from  1  to  2  is 


121- 


E10 


or 


Each  equation  of  the  form  (18a)  or  (186)  is  in  reality  equiv- 
alent to  two  equations,  since  the  sum  of  all  the  "  real "  terms 
on  one  side  must  be  equal  to  the  sum  of  all  the  real  terms  on 
the  other  side,  and  similarly,  the  sum  of  all  the  "  j  "  terms  on  one 
side  must  be  equal  to  the  sum  of  all  the  "  /  "  terms  on  the  other 


SYMBOLIC  METHOD  385 

side;  the  denominators  of  all  fractions  having  been  cleared  of 
"  /"  terms  by  the  transformation  given  by  equation  (17).  This 
is  merely  another  way  of  stating  the  fact  that  the  component  in 
any  direction  of  the  resultant  of  any  number  of  vectors  must  be 
equal  to  the  algebraic  sum  of  the  components  in  this  direction 
of  all  the  individual  vectors.  Applying  these  two  laws  to  any 
network  enables  one,  therefore,  to  calculate  both  components  of 
every  p.d.  and  every  current,  when  the  impedances  and  the 
electromotive  forces  are  known. 

It  should  be  clearly  borne  in  mind  that  the  above  equations 
are  true  only  when  the  currents,  the  e.m.f.'s,  and  the  impedances 
are  expressed  as  vector  quantities.  These  equations  are  not 
true  when  the  numerical  values  of  these  quantities  are  employed. 
The  indications  of  alternating  current  voltmeters  and  voltmeters 
give  only  the  effective  values  of  currents  and  p.  d.'s  respectively, 
they  do  not  indicate  their  phase  relations.  Hence,  the  sum  of  the 
currents  entering  a  junction  as  measured  by  ammeters  in  the  vari- 
ous branches  is  not  necessarily  zero;  again,  the  sum  of  the  potential 
drops  in  the  various  branches  of  a  closed  loop,  as  measured  by  volt- 
meters in  the  various  branches,  is  not  necessarily  zero. 

Again,  these  equations  hold  only  when  the  currents  and  e.m.f.'s 
are  all  simple  harmonic  functions  of  the  same  frequency  and  the 
resistances  and  reactances  are  constant.  When,  however,  the  re- 
sistances, inductances  and  capacities  are  constant,  a  similar  set 
of  equations  holds  for  each  frequency  that  may  be  present.  Since 
the  equations  are  all  linear  in  the  I's  and  E's,  the  currents  and 
e.m.f.'s  of  any  given  frequency  will  be  uninfluenced  by  the  pres- 
ence of  currents  or  e.m.f.'s  of  any  other  frequency.  Hence,  when 
the  harmonics  present  in  each  e.m.f.  are  known,  the  harmonics 
present  in  each  current  may  be  calculated  by  solving  the  equa- 
tions corresponding  to  the  frequency  of  this  particular  harmonic, 
these  equations  being  exactly  the  same  as  would  hold  were  all  the 
other  harmonics  absent. 

Note  particularly  that  the  above  equations  do  not  hold  for 
transient  currents;  they  apply  only  after  the  transient  terms 
(see  Article  201)  have  become  zero. 

215.  Expression  for  Average  Power  in  Symbolic  Notation.  — 
When  the  current  in  any  circuit  is  i  =  \/2I  sin  (a)t  +  0)  and  the 
potential  drop  in  the  circuit  (in  the  direction  of  the  current)  is 


386  ELECTRICAL  ENGINEERING 

v=\/2  V  sin  (tot+9f)  the  average  power  input  into  the  circuit  is 

P  =  VI  cos  (9'  -6) 

where  V  and  /  are  the  effective  values  of  the  current  and  p.d. 
(see  equation  9  of  Chapter  VII)  .  The  current  i  =\/2  7  sin(a)  t  +  0} 
may  also  be  written 

i  =  V2  /  cos  6  sin  a)t  +  \/2  /  sin  9  cos  oj  t 
and  may  therefore  be  represented  by  the  vector 

/=/,+//, 

where 

/!  =1  COS  9 

1  2=I  sin  9 
Similarly,  the  p.d.  v  =\/2  V  sin  (a)t  +  0f)  may  be  written 

v  =\/2  V  cos  9'  sin  a)t  +  V2  V  sin  9'  cos  CD  t 
and  may  therefore  be  represented  by  the  vector 


where 

V,  =  V  cos9l 
V^VsinO, 
Expanding  the  power  equation  we  get 

P  =  VI  cos  9  cos  9'  +  VI  sin  9  sin  9f 

=  Vcos0'Xlcos0+V  sin  0'Xl  sin  9 
But  V  cos  9'  =  Vl\  I  cos  9  =/!,-  V  sin  0'  =  72;  I  sin  9  =72. 
Hence    the    average    power    corresponding    to    7=7x  +  /72    and 
7  =  7^/7,  is 

P  =  VJ1+V2I2  (19) 

Note  that  this  expression  is  not  equal  to  the  real  part  of  the  product 
of  V  =  Vl  +  jV2  and  7=71  +  /72,  when  "/"  is  considered  equiv- 
alent to  V—  1,  which  product  gives 

F7  =  7171-7272  +  /(^172+F271). 

The  average  power  is  the  real  part  of  this  product  with  the  sign 
between  the  two  terms  7^  and  7272  reversed. 

216.   Expression  for  Reactive  Power  in  Symbolic  Notation.  - 
The  reactive  power  corresponding  to  the  current  i=\/2Isin(ajt  +  9) 
and  the  p.d.  v  =  V2  7  sin  (  cot  +  9')  is  (see  Article  184) 

U  =  VIsin(9'-9] 

(Note  that  the  reactive   power  is  taken  as  positive  when  the 
current  lags  behind  the  p.d.,  i.e.,  when 
Expanding  the  equation  for  U  we  get 


SYMBOLIC  METHOD  387 

U  =  VI  sin  Q'  cos  9  -VI  cos  6f  sin  0 

=  V  sin  0'  I  cos  9-  V  cos  0'  I  sin  0 
Therefore 

ff-FA-FA  (20) 

When  /!,  Vl}  I2  and  V2  have  the  same  signification  as  in  the 
preceding  article.  Therefore  the  reactive  power  is  not  equal  to 
the  "/"  part  of  the  product  V  =  V,-\-jVz  and  I^I^jV^  when 
/  is  considered  equivalent  to  V—  1,  but  is  the  "  /  "  part  of  this 
product  with  the  sign  between  the  terms  VJ2  and  V2Ii  reversed. 

217.  Expression  for  Power  Factor  in  Symbolic  Notation.  — 
The  power  factor  corresponding  to  the  current  /  =Il  -f  jI2  and  the 
p.d.  V  =  Vl  +  jV2  is  equal  to  the  ratio  of  the  average  power 
P  =  VJi  +  V2I2  to  the  product  of  the  effect  current  /  and  the 
effective  p.d.  V,  that  is 

ftj.-Jfc-  V^+V^  (21) 

VI 


The  power-factor  angle,  i.e.,  the  angle  the  cosine  of  which  is  equal 
to  the  power  factor,  is  6f  —  0,  since  &  —  6  is  the  difference  in  phase 
between  current  and  p.d.  Therefore  from  (19)  and  (20) 

[V  1  —  V  T  ~i 
(21a) 
FA+FAJ 

218.  Examples  of  the  Use  of  the  Symbolic  Method.  —  Problem 
I,  Series  Circuits.  —  •  An  impedance  zl  has  a  resistance  of  3  ohms 
and  an  inductive  reactance  of  4  ohms;  a  second  impedance  z2  has 


^-rAA/WV- 

V*  V,,. 

Fig.  122. 

a  resistance  of  8  ohms  and  a  capacity  reactance  of  6  ohms.  2t 
is  then  represented  symbolically  as 

and  z2  as 

Let  these  two  impedances  be  connected  in  series,  Fig.  122,  and 
let  an  e.  m.  f.  of  100  volts  be  impressed  across  them  from  a  to  c. 
Choosing  the  vector  representing  the  potential  drop  from  a  to  c 


388 


ELECTRICAL  ENGINEERING 


as  the  axis  of  reference,  and  calling  the  current  from  a  to  c  Ic 
we  have,  that  (3+/4  +  S-/6)  Iac  =  100+/0  whence 

100       1100+/200 

Lnr  = = =o.o  +  ?l  6 

11-2/        121  +  4 

Hence  the  effective  value  of  the  current  is 

7  =  l/(8.8)2+(1.6)2=8.94  amperes 
and  it  leads  the  potential  drop  from  a  to  c  by  the  angle 

tan'1— =10.3° 

8.8 

The  potential  drop  across  the  first  impedance  is 

Ya&=(3+/4)  (8.8  +  /1.6)=26.4-6.4  +  /  (35.2  +  4.8) 

=20+/40 
which  has  the  effective  value 

7a6=l/(20)2+(40)2=44.7  volts 
and  leads  the  potential  drop  Vac  by  the  angle 

40 

tan1  _=63.5° 
20 


Vbo=894 
Fig.  123. 

The  potential  drop  across  the  second  impedance  is 

76c=(8-/6)  (8.8  +  /1.6)=70.4  +  9.6-y  (52.8-1.28) 

=80  -/40 
which  has  the  effective  value 

76c=l/(80)2+(40)2=89.4  volts 
and  lags  behind  the  potential  drop  Vac  by  the  angle 


SYMBOLIC  METHOD 


389 


4  I40 

tan  ~*  — 
80 


=26.5C 


The  complete  vector  diagram  is  given  in  Fig.  123.     Note  that  the 
current  is  plotted  to  ten  times  the  scale  of  the  p.d. 

The  power  input  into  the  first  impedance  is 

Wab  =8.8X20  +  1.6X40=240  watts 

The  power  input  into  the  second  impedance  is 

Wbc  =8.8  X  80 -  1 .6 X 40  =640  watts 
The  total  power  input  is 

Wac  =8.8X100  +  1.6X0  =880  watts 
which  of  course  is  the  sum  of  Wab  and  W^. 
Problem  2.     Parallel  Circuits. 

Next,  let  these  two  impedances  be  connected  in  parallel,  Fig. 
124,  and  let  the  total  current  taken  by  the  two  be  100  amperes. 

Choosing  the  vector  representing  the  total  current  from  a  to 
6  as  the  axis  of  reference,  and  calling  I'ab  =//  +  ///  the  current 
in  No.  1  from  a  to  b  and  I"ab  =//'  +  ///'  the  current  in  No.  2 
from  a  to  6,  we  have  from  equations  (18)  that 

IS  +  jI*'  +  If  +  jIf  =  100  +  /0 
and 

(3  +  /4)  (//  +  ///)- (8 -/6)  (//'  +  ///)  =0 
Whence 

//  +  If  +  /  (/,'  +  If)  =  100  +  j  0 
37/  -  4/2'  -  /  (4//  +  3//)  =8//'  +  67/  +  /  (81  f  -  6//0 


Equating  the  real  and  the  "/  "  terms  in  these  two  equations  we  get 
//  +  //'  =  100 


3/i'_4/2'=8/1" 


390 


ELECTRICAL  ENGINEERING 


Solving  these  equations  we  get 

7/=80 

72'  =  -40 

7/'=20 

72"=40 
Whence  the  current  in  No.  1  from  a  to  b  is 

7'a6=80-/40 
which  has  the  effective  value 

/'=1/(80)2+(40)2=89.4  amperes 
and  lags  behind  the  total  current  Iab  by  the  angle 

tan-1  —  =26.5° 
80 

The  current  in  No.  2  from  a  to  b  is 


which  has  the  effective  value 

7"  =  \/(20)2+(40)2=44.7  amperes 
and  leads  the  total  current  Iab  by  the  angle 

tan'1  ^=63.5° 
20 

The  potential  drop  across  each  impedance  in  the  direction  from 
a  to  b  is 

7o6=(3  +  /4)  (80-/40)  =240  +  160  +  /  (320-120) 


which  has  the  effective  value 

V=  V(400)2  +  (200)2=447  volts 
and  leads  the  total  current  by  the  angle 

/an-200  =26.5 
400 


I' =89.4 
Fig.  125. 


(Note;  this  problem  can  be  solved  more  simply  by  calculating 


SYMBOLIC  METHOD  391 

the  admittances  corresponding  to  the  two  impedances  by  equa- 
tions (37)  of  Chapter  VII,  and  then  proceeding  in  an  entirely 
analogous  manner  to  that  employed  in  Problem  1.  The  method 
above  given,  however,  illustrates  the  manner  in  which  any  problem 
concerning  a  network  of  circuits  may  be  attacked.) 

The  vector  diagram  for  the  currents  and  p.d.  is  given  in  Fig. 
125. 

The  p.d.  is  plotted  to  one-tenth  the  scale  of  the  current. 
The  power  input  into  No.  1  is 

W  =80  X  400  -  40  X  200  =24,000  watts 
The  power  input  into  No.  2  is 

W"  =20  X  400  +  40  X  200  =  16,000  watts 
The  total  power  input  is 

W  =  100  X  400  +  0  X  200  =40,000  watts 

SUMMARY  OF  IMPORTANT  DEFINITIONS  AND 

PRINCIPLES 

1.  A  vector  of  length  A  making  an  angle  9  with  any  arbi- 
trarily chosen  axis  of  reference  may  be  represented  symbolically 
by  the  expression 

A=Al+jA, 

where  Al  is  the  component  (  =  A  cos  6}  of  the  vector  A  along  the 
axis  and  A2  is  the  component  (  =  A  sin  6  )  of  the  vector  A  making 
a  positive  angle  of  90°  (counter-clockwise)  with  the  axis.  The 
symbol  "  /  "  indicates  that  the  component  A2  makes  the  angle 
90°  with  the  axis. 


2.  The  length  of  any  vector  Al  +  jA2  is  A=v/A12  +  ^22  and 

^ 

the  angle  which  it  makes  with  the  axis  is  6  =tanl—         * 

A, 

3.  The  sum  of  two  vectors  At  +jA2  and  A,f  +jA2'  referred  to 
the  same  axis  is 


4.  In  addition  to  showing  direction  the  symbol  j  also  has  the 
following  properties: 

(a)  Multiplying  a  vector  by  /  rotates  the  vector  through  90° 
in  the  positive  direction. 

(6)  ii=f—i. 

(c)   jK  =  Kj  when  K  is  a  constant,  real  or  complex. 
(e)   jX±Xj  when  X  is  a  rotating  vector. 


392  ELECTRICAL  ENGINEERING 

(d)  jX  =  --  ;   when  X  is  a  vector  rotating  with  a  constant 
CD     dt 

angular  velocity  CD. 

5.  The  symbolic  notations  for  a  sine-wave  current,  a  sine-wave 
p.d.,  and  a  sine-wave  e.  m.  /.  are  respectively 


where  the  two  components  of  each  quantity  are  any  two  compo- 
nents at  right  angles  to  each  other  and  each  component  is  a  vector 
rotating  with  an  angular  velocity  aj  =2  IT  f,  where  /  is  the  fre- 
quency. 

6.  The  symbolic  notation  for  an  impedance  of  constant  resist- 
ance r  and  constant  reactance  x  to  a  sine-wave  current  is  the 
complex  number 


7.  The  symbolic  notation  for  an  admittance  of  constant  con- 
ductance g  and  constant  susceptance  b  to  a  sine-wave  current  is 
the  complex  number 

y=g-jb 

8.  The  symbolic  notation  for  the  p.d.  V  in  an  impedance  z 
due  to  a  current  /  is 

V=zl 

9.  The  symbolic  notation  for  the  current  I  in  an  admittance 
y  due  to  a  p.d.  V  is 

I=yV 

10.  The  relation  between  impedance  and  admittance  in  sym- 
bolic notation  is 

1 
y  — 

11.  The  symbolic  notation  for  the  equivalent  impedance  Z  of 
two  impedances  zl  and  z2  in  parallel  is 


Z,  2, 


12.  An  expression  of  the  form  where  at   and  a2  are 

a,  ±  ]a2 

constants  is  equal  to 


SYMBOLIC  METHOD  393 


13.  KirchhofFs  Laws  in  symbolic  notation  are 


where  the  currents  7  and  the  electromotive  forces  E  are  all  referred 
to  the  same  axis  of  reference. 

14.  The  average  power  corresponding  to  a  p.d.  V  =  V1-\-jV2  and 
a  current  7=71+/72,  both  referred  to  the  same  axis  of  reference  is 

and  the  power  factor  is 


VI 


PROBLEMS 

1.  A  vector  A  10  units  in  length  makes  an  angle  of  60°  in  a 
counter-clockwise  direction  with  another  vector   B  15  units  in 
length.     Find  the  symbolic  expressions  of  (1)  A  and  B  respec- 
tively,   and  (2)  A  +  B,  and  (3)  A  -B  referred  to  B  as  the  axis 
of  reference  in  each  case. 

Ans.:  (1)  5 +  / 8.66  and  10;  (2)  20 +  / 8.66;  (3)  -10  +  /8.66. 

2.  A  certain  vector  is  represented  symbolically  by  the  expres- 
sion 12  +  /24.      Find  the  symbolic  expression  of  a  vector  equal 
in  length  (1)  when  it  makes  an  angle  of  —  40°  with  the  axis  of 
reference,  and  (2)  when  it  makes  a  right  angle  with  the  axis  of 
reference  in  the  clockwise  direction. 

Ans.:  (1)  20.6-/17.27;  (2)  -/26.8. 

3.  Three  impedances  A,  B  and  C  have  resistances  of  5,  8  and 
3  ohms  respectively  and  reactances  of  14,  0  and  —10  ohms  respec- 
tively.    Find  the  symbolic  expressions  of  the  resultant  impedance 
of  these  three  impedances,  (1)  when  they  are  connected  in  series, 
(2)  when  they  are  connected  in  parallel,  and  (3)  find  the  symbolic 
expression  of  the  resultant  admittance  of  these  three  impedances 
when  they  are  connected  in  parallel. 

Ans.:  (1)  16  +  /4;  (2)  5.56-/0.902;  (3)  0.1751 +/0.0284. 

4.  An  e.  m.  /.  of  150  volts  and  25  cycles  is  impressed  upon  a 
series  circuit  consisting  of  a  resistance  of  5  ohms,  an  inductance  of 
0.05  henry  and  a  capacity  of  1000  microfarads.     If  the  vector 


394  ELECTRICAL  ENGINEERING 

representing  the  impressed  e.  m.  f.  is  taken  as  the  axis  of  reference 
find(l)  the  symbolic  expression  for  the  current  flowing  in  the  circuit, 
and  (2)  the  average  rate  at  which  energy  is  given  to  the  circuit. 
Ans.:  (1)  27.6-/ 8.17;  (2)  4140  watts. 

5.  Two  impedances   A  and  B  of  4  and  12  ohms  resistance 
respectively  and  15  and  —8  ohms  reactance  respectively  are  con- 
nected in  series.     If  the  current  established  in  this  circuit  is  10 
amperes,  find  the  symbolic  expression  for  the  potential  drop,  (1) 
through  A,  (2)  through  B,  and  (3)  through  the  entire  circuit,  all 
referred  to  the  current  as  the  axis  of  reference. 

Ans.:  (1)  40  +  /150;  (2)  120-^80;  (3)  160  +  /70. 

6.  If  the  two  impedances  described  in  Problem  5  are  connected 
in  parallel,  and  the  current  in  A  is  5  amperes,  find  the  symbolic 
expressions  for  the  currents  in  A  and  B  respectively  referred  to 
the  potential  drop  across  the  parallel  circuit  as  the  axis  of  reference. 

Ans.:  1.289-/4.84  and  4.48  +  /2.9S. 

7.  When  a  given  60- cycle  e.  m.  f.  is  impressed  upon  a  series 
circuit  of  5  ohms  resistance,  0.1  henry  inductance  and  50  microfarads 
capacity,  the  symbolic  expression  for  the  'fundamental  of  the  cur- 
rent is  12  —  /1 0  and  for  the  third  harmonic  of  the  current  6  +  /8 
referred  respectively  to  the  first  and  third  harmonics  respectively 
of  the  impressed  e.  m.  /.     Find(l)  the  effective  value  of  the  im- 
pressed e.  m.  f.  and  (2)  the  average  power  dissipated  as  heat  energy 
in  the  circuit. 

Ans.:  (1)  988  volts;  (2)  1720  watts. 

8.  An  impedance  A  is  connected  in  series  with  two  impedances 
B  and  C  connected  in  parallel.     The  resistances  of  A,  B  and  C 
are  3,  5  and  6  ohms  respectively  and  the  reactances  are  5,  —  7  and 
3  ohms  respectively.     If  an  e.  m.  /.  of  200  volts  is  impressed  across 
the  entire  circuit,  find  the  symbolic  expression  of  the  currents  in 
A,  B  and  C  referred  to  the  potential  drop  in  A. 

Ans.:  11.4-/19.0;  12.4-/2.75;  -1.01  -/16.3. 

9.  An  e.  m.  f.  of  100  volts  is  impressed  upon  a  series  circuit 
formed  by  two  impedances  A  and  B  in  series.     The  impedance  A 
has  a  resistance  of  4  ohms  and  a  reactance  of  6  ohms  and  the 
potential  drop  across  A  is  100  volts.     If  the  entire  circuit  absorbs 
1  kilowatt  of  power,  find  the  symbolic  expression  of  (1)  the  cur- 
rent, (2)  the  potential  drop  through  A,  and  (3)  the  potential  drop 
through  B}  all  referred  to  the  e.  m.  /.  impressed  upon  the  entire 
circuit. 


SYMBOLIC  METHOD  395 

Ans.:  (l)10±/9.58.;(2)-17.5  +  /98.3or  97.5  +  /21.7;(3)  117.5 
-/98.3  or  2.5  -j2l. 7. 

10.  Two  impedances  A  and  B  are  connected  in  parallel.     The 
respective  symbolic  expressions  for  the  impedances  of  A  and  B 
are  4  +  /8  and  5  —  /3.     If  the  total  current  supplied  to  this  parallel 
circuit  is  20  amperes,  find  the  symbolic  expressions  for  (1)  the 
currents  in  A  and  B,  and  (2)  the  potential  drop  across  the  parallel 
circuit,  all  referred  to  the  total  current  supplied  to  the  circuit. 

Ans.:  (1)  5.66-/9.81  and  14.34+3*9.81 ;  (2)  101.1 +/6.0. 

11.  A  500- volt,  60-cycle  alternator  is  delivering  25  kw.  at  a 
power  factor  of  85%,  lagging  current.     The  armature  has  a  resis- 
tance of  0.25  ohm  and  an  inductance  of  0.001  henry.     Referring  all 
vectors  to  the  terminal  voltage  of  the  alternator,  find  the  symbolic 
expression  for  (1)  the  armature  current,  (2)  the  armature  voltage, 
and  (3)  the  resistance  drop  in  the  armature. 

Ans.:  (1)  50.0-/31.0;    (2)  524  +  jll.l ;  (3)  12.5-/7.75. 

12.  The  generator  in  Problem  11  supplies  power  to  a  motor 
over  a  line  of  0.3  ohm  resistance  and  +0.6  ohm  reactance.     If  the 
armature  of  the  motor  has  a  resistance  of  0.2  ohm  and  a  reactance 
of  0.3  ohm,  find  the  symbolic  expression  of  the  armature  voltage 
of  the  motor  referred  to  the  terminal  voltage  of  the  motor. 

Ans.:  -448.1  +/9.7. 


IX 
THREE-PHASE  ALTERNATING  CURRENTS 

219.  Polyphase  Alternating  Currents.  —  A  polyphase  alterna- 
tor is  an  alternator  upon  the  armature  core  of  which  are  wound 
two  or  more  independent  windings  which  are  arranged  with  respect 
to  each  other  in  such  a  manner  that  the  electromotive  forces  in 
the  various  windings  differ  in  phase  by  a  constant  angle.  In  a 
two-phase  alternator  there  are  two  independent  windings,  which 
are  arranged  in  such  a  manner  that  when  the  e.  m.  f.  induced  in  one 
winding  is  a  maximum,  the  e.  m.  /.  induced  in  the  other  winding 
is  zero,  that  is,  the  e.  m.  f.  induced  in  one  winding  is  in  quad- 
rature with  the  e.  m.  f.  induced  in  the  other  winding ;  this  effect 
is  obtained  by  placing  the  two  windings  on  the  core  in  such 
a  manner  that  when  a  group  of  conductors  in  one  winding  is 
directly  under  a  north  pole,  the  corresponding  group  of  con- 
ductors in  the  second  winding  is  midway  between  a  north  pole 
and  a  south  pole.  In  the  case  of  a  two-pole  machine  this  means 
that  a  group  of  conductors  in  one  winding  is  placed  90  degrees 
ahead  of  the  corresponding  group  in  the  other  winding ;  similarly, 
in  the  case  of  a  multipolar  machine,  if  we  call  the  distance  between 
two  successive  poles  of  like  sign  equal  to  360  "  electrical  "  degrees, 
we  may  describe  the  relative  position  of  the  two  windings  as  such 
that  the  corresponding  conductors  in  the  two  windings  are  spaced 
90  electrical  degrees  apart.  In  general  the  two  windings  overlap 
each  other,  or  are  "  distributed  "  over  the  armature  surface,  but 
for  each  conductor  in  one  winding  there  is  a  corresponding  con- 
ductor in  the  other  winding  90  electrical  degrees,  or  one  quarter 
of  the  pole  pitch,  ahead  of  the  conductor  in  the  first  winding. 

A  two-phase  alternator  may  be  provided  with  a  separate  pair 
of  slip  rings  for  each  winding,  that  is,  four  rings  in  all ;  or  with 
three  slip  rings,  and  one  of  these  rings  made  to  serve  as  a  common 
return  for  the  two  windings. 

A  three-phase  alternator  is  similar  in  construction  to  a  two- 
phase  machine  except  that  three  separate  windings  are  employed, 
the  corresponding  conductors  in  the  three  windings  being  spaced 

396 


THREE-PHASE  ALTERNATING  CURRENTS  397 

120  electrical  degrees  apart,  and  the 
induced  electromotive  forces  therefore 
differ  in  phase  by  120°.  These  wind- 
ings are  connected  electrically  in  two 
different  ways.  The  three  windings 
may  be  connected  end  to  end  as  shown 
in  Fig.  126,  and  leads  from  the  junc- 
tions 1,  2  and  3,  brought  out  to 
three  slip  rings ;  or  one  end  of  each  of 
the  three  windings  may  be  connected 

?  /  A  Connection 

to  a  common  junction,  called  the 
"  neutral  point/'  as  shown  in  Fig. 

127,  and  the  other  ends  1,  2  and  3  brought  out  to  three  slip  rings. 
In  some  cases  the  common  junction  or  neutral  point  is  also  con- 
nected to  a  fourth  slip  ring.  A  three-phase  alternator  with  its 
windings  connected  as  shown  in  Fig.  126  is  said  to  be  "  mesh  " 
connected,  the  latter  from  the  Greek  capital  letter 
3  "  A. "  When  the  windings  are  con- 

nected as  shown  in  Fig.  128  the  alter- 
nator is  said  to  be  "  star  "  or  "  Y  " 
connected.     In   large   generators    the 
armature  is  stationary  and  the  field 
rotates;  in  such  a  case  the  armature 
windings  are  connected  to  fixed  ter- 
minals and  the  two  ends  of  the  field 
1  winding    are    connected    to    two    slip 
Fig-  127-  rings.      Continuous  current  is  supplied 

to  the  field  through  brushes  bearing  on  these  two  slip  rings. 
The  advantages  in  using  three-phase  currents  are  chiefly : 

1.  The  more  satisfactory  operation  of  three-phase  motors  as 
compared  with  single-phase  machines. 

2.  The  saving  in  the  amount  of  copper  (25  per  cent)  as  com- 
pared with  a  single-phase  system  for  the  same  voltage  between 
wires. 

3.  The  lesser  cost  of  three-phase  generators  and  motors  of  the 
same  power  and  for  the  same  voltage  between  terminals. 

4.  Better  voltage  regulation  of  three-phase  generators. 

220.  Vector  Sum  of  the  Induced  E.  M.  F.'s  in  the  Three  Wind- 
ings of  a  Three-Phase  Generator  Equals  Zero.  —  Let  Eia,  E23 
and  E3l,  Fig.  128,  be  the  three  e.m.f.'s  induced  in  the  three  wind- 


398 


ELECTRICAL  ENGINEERING 


ings  of  a  three-phase  generator,  taken  as  positive  in  the  counter-- 
clockwise direction  around  the  closed  loop  formed  by  the  windings  of 


Fig.  128. 

the  generator.  In  a  properly  designed  machine  these  e.m.f.'s  have 
equal  effective  values,  but  due  to  the  relative  position  of  the  wind- 
ings they  differ  in  phase  by  120°;  that  is  E12  leads  E23  by  120°  and 
Eal  leads  E12  by  120°.  Hence  the  symbolic  expressions  for  the 
three  e.m.f.'s  referred  to  E12  as  the  line  of  reference  are,  when  these 
e.m.f.'s.  are  sine  waves, 

E=        1+     0      E 


(1) 


.„=(-!-,%„ 


and  therefore 


Hence  the  total  e.m.f.  at  any  instant  acting  around  the  closed 
loop  formed  by  the  three  windings  is  zero,  and  therefore  when  no 
external  circuit  is  connected  to  the  three  terminals,  i.e.,  when 
the  generator  is  running  on  "  open  "  (external)  circuit,  there  will 
be  no  flow  of  current  in  the  armature  provided  the  e.m.f.'s  are 
sine  waves. 

Similarly,  in  the  case  of  a  three-phase  F-connected  generator, 
Fig.  129,  the  induced  e.m.f  '.'s  EQl,  EQ2,  and  E^  are  all  equal  in 
effective  value  and  differ  in  phase  by  120°.  Hence  choosing  Eol  as 
the  line  of  reference  and  taking  these  e.m.f.'s  as  positive  in  the 


THREE-PHASE  ALTERNATING  CURRENTS 


399 


direction  away  from  the  neutral  point,  we  have,  when  the  e.m.f.'s 
are  sine  waves,  that 


(2) 


and  therefore 

#01  +  #02+#03=0 

When  the  generator  is  running  on  open  (external)  circuit  there 


Fig.    129. 

is  of  course  no  current  in  the  coils,  since  each  coil  is  open  at  one 
end. 

221.  Relation  Between  Coil  E.M.F.  and  E.M.F.  Between 
Terminals  in  a  Y-Connected  Three-Phase  Generator.  —The  e.m.f. 
between  any  pair  of  terminals,  or  the  equivalent  "  A  "  e.m.f. 
of  a  F-connected  generator  is  equal  to  the  vector  sum  of  the 
e.m.f.'s  in  the  two  coils  in  the  direction  from  one  terminal  to  the 
other.  Using  the  above  notation,  and  assuming  sine-wave 
e.m.f.'s,  we  have  that  the  e.m.f.  between  the  terminals  2  and  3  in 
the  F-connected  generator  in  the  direction  from  2  to  3  is 


E23  = 


E 


Similarly  for  the  e.  m.  f.  's  between  the  terminals  3  and   1   and 


400 


ELECTRICAL  ENGINEERING 


between  the  terminals  1   and  2;   hence,   the  equivalent  "A"  e. 
m.  f.'s,  that  is,  the  e.m.f.'s  between  terminals,  are 


(3) 


-j  V3 


77F       „  A         /O       77* 

That  is,  the  equivalent  A  e.m.f.  is  equal  numerically  to  the  square 
3  root  of  three  times  the  effective  value  of 

the  Y  e.  m.  f.  and  lags  90°  behind  the 
e.m.f.  in  the  opposite  branch  of  the  Y. 
The  A  e.m.f. 's  are  therefore  represented 
by  the  sides  of  the  triangle  formed  by 
connecting  the  ends  of  the  three  vectors 
representing  the  Y  e.  m.  f.'s,  the  positive 
direction  being  taken  in  the  counter- 
clockwise direction,  see  Fig.  130. 

222.  Currents  from  a  Three-Phase 
Generator.  —  Generator  and  Load  Both 
Y-Connected.  — When  the  three  terminals  of  a  three-phase  genera- 
tor, either  A  or  Y  connected,  are  connected  to  either  a  Y  or  A 
connected  load,  currents  equal  in  effective  value  and  differing  in 
phase  by  120°  will  flow  from  each  terminal,  provided  (1)  that  the 
resistance  and  reactances  of  the  generator  windings  are  respec- 
tively the  same  for  the  three  windings,  (2)  that  the  resistances  and 
reactances  of  the  windings  of  the  receiving  device  or  load  are 
respectively  the  same  for  the  three  windings,  (3)  that  the  e.  m.  f.'s 
in  each  winding  of  the  generator  are  equal  in  effective  value  and 
differ  in  phase  by  120°,  and  (4)  that  the  back  e.  m.  f.'s,  if  any,  in 
the  windings  of  the  load  are  equal  in  effective  value  and  differ  in 
phase  by  120°.  When  the  e.  m.  f.'s  in  each  phase  of  a  three-phase 
system  are  equal  in  effective  value  and  differ  in  phase  by  120° 
and  the  currents  in  each  phase  are  equal  in  effective  value  and 
differ  in  phase  by  120°,  the  system  is  said  to  be  "  balanced." 

Consider  the  simple  case  of  a  F-connected  generator  to 
the  three  terminals  of  which  are  connected  three  equal  im- 
pedances in  F,  Fig.  131.  Let  z  represent  the  total  impedance 
of  each  winding  of  the  load  plus,  vectorially,  the  impedance  of 
the  generator  winding  in  series  with  it.  For  the  closed  loops 
OlaO'620  and  O260'c30,  we  have,  assuming  sine-wave  currents  and 
e.m.f. 's,  that 


THREE-PHASE  ALTERNATING  CURRENTS  401 

E01-E02=zIol-zI02  (a) 

Also  at  the  junction  0  or  0'  we  have  that 

which  give  three  equations  in  the  three  unknown  quantities 
701  /02  and  703.  Referring  the  e.m.f.'s  to  E01  as  the  axis  of  refer- 
ence, equations  (a)  and  (b)  may  be  written 


Fig.  131. 


(d) 


Substituting  the  value  of  703  from  equation  (c)  in  the  second  of 
these  equations,  gives 


which  subtracted  from  (d)  gives 
/3_  .3V  3 

Therefore 


Similarly  for  the  other  two  phases;  hence 

^701  =£'01 

2/02  =  E02 

z703  =  E03 


(4) 


402  ELECTRICAL  ENGINEERING 

Consequently,  since  the  three  e.m.f.'s  are  equal  in  effective 
value  and  differ  in  phase  by  120°,  the  currents  in  the  three  im- 
pedances have  equal  effective  values  and  differ  in  phase  by  120°. 
These  currents,  however,  are  not  'in  phase  with  the  corresponding 
e.m.f.'s  except  in  the  special  case  when  the  reactance  component 
of  the  impedance  is  zero. 

When  the  load  is  a  motor  or  other  device  developing  in  the 
three  windings  back  e.m.f.'s  equal  in  effective  value  and  differing 
in  phase  by  120°,  it  can  readily  be  shown  that  equations  (4) 
still  hold  provided  the  E's  in  these  equations  are  taken  to  repre- 
sent the  vector  difference  of  the  generator  e.m.f.  per  winding  and 
the  back  e.m.f.  of  the  load  per  winding. 

From  the  above  discussion  it  therefore  follows  that  when  a 
balanced  F-connected  load  is  connected  to  a  F-connected  gener- 
ator, the  current  established  in  each  winding  of  the  load  is  exactly 
the  same  as  would  be  produced  were  this  winding  connected  by 
itself  to  a  single-phase  generator  having  an  e.  m.  f.  equal  to  the 
e.  m.  f.  induced  in  each  winding  of  the  generator  and  an  internal 
impedance  equal  to  the  impedance  of  each  of  these  windings.  It 
also  follows  from  equation  (4)  that  the  neutral  point  of  the  gener- 
ator and  the  neutral  point  of  the  load  must  be  at  the  same  poten- 
tial, since  the  rise  of  potential  in  the  generator  from  the  neutral 
point  to  any  one  slip  ring  due  to  the  induced  e.  m.  f.  is  equal  to 
the  drop  in  potential  zl  due  to  the  resultant  impedance  of  the 
generator,  line  and  load.  Consequently,  were  the  neutral  points 
of  the  generator  and  load  connected  by  a  wire,  no  current  would 
flow  in  this  "  neutral "  wire,  provided  the  system  is  balanced.  In 
case  the  load  is  not  balanced,  a  current  would  flow  in  the  neutral 
wire ;  such  a  wire  is  sometimes  installed  to  reduce  the  voltage  drop 
between  generator  and  load  in  case  of  unbalancing.  In  general, 
however,  a  three-phase  system  is  very  nearly  balanced,  as  three- 
phase  motors  and  rotary  converters  are  designed  to  take  the  same 
current  per  phase,  and  in  the  case  of  a  lamp  load  or  single-phase 
motors  used  on  a  three-phase  system,  care  is  taken  to  distribute  the 
various  lamps  and  single-phase  motors  equally  on  the  three-phases. 

223.  Currents  from  a  Three-Phase  Generator. — Generator  A- 
Connected  and  Load  Y-Connected.  —  When  the  generator  is  A- 
connected  the  induced  e.m.f.'s  between  terminals  are  E12,  E23 
and  E3l.  Hence,  neglecting  the  resistance  and  reactance  of  the 
generator,  and  assuming  sine-wave  e.m.f.'s  and  currents,  we  have 


THREE-PHASE  ALTERNATING  CURRENTS  403 


E23=zln3-In2)  (6) 

/OI  +  /o2  +  /o3=0  (c) 

Referring  E12  and  E23  to  E12  as  the  line  of  reference  and  substitut- 
ing in  (a)  and  (6)  the  value  of  702  from  equation  (c),  we  have 


Multiplying  the  second  equation  by  2  and  adding  it  to  the  first  gives 

jV3E12=3z!03 

or 

zl    -lEl2 

•"OS  — — '—= 

\/3 
Similarly  for  the  other  two  phases ;  whence 


V3 

:?i      .  (5) 


V3 

Hence,  since  the  three  A  e.m.f.'s  are  equal  in  effective  value  and 
differ  in  phase  by  120°,  the  currents  in  the  three  impedances  are 
equal  in  effective  value  and  differ  120°  in  phase.  Note  that  the 

*  T?          'Jf  'Tf 

vectors  ^^,  l^L  and  i_£!  are  equal  to  the  vectors  represented 

\/3   V3  V3 

by  the  lines  drawn  from  the  center  of  the  triangle  formed  by  the 
vectors  E12,  E23  and  E3l-y  hence  the  currents  taken  from  slip  rings 
of  a  A-connected  generator  by  a  balanced  load  are,  neglecting 
armature  impedance,  exactly  the  same  as  would  be  produced 
by  a  F-connected  generator  developing  e.m.f.'s  Eol,  E02,  En3 
per  winding  where 


Compare  with  equations  (3).     A  A-connected  generator,  when  sup- 


404 


ELECTRICAL  ENGINEERING 


plying  a  balanced  load  is  therefore  equivalent  to  a  Y -connected  genera- 
te 
tor  developing  an  e.m.J.  per  winding  Ey  numerically  equal  to ^, 

V3 

where  E^  is  the  e.m.f.  per  winding  of  the  ^-connected  generator. 
As  will  be  shown  presently,  when  the  impedance  of  the  generator 
is  taken  into  account,  as  it  must  be  in  all  practical  problems,  it 
is  also  necessary,  in  order  that  the  F-connected  and  the  A-con- 
nected  generator  produce  the  same  current  in  the  external  circuit, 

that  the  impedance  per 
winding  of  the  equivalent 
F-connected  generator  be 
equal  to  one-third  the  im- 
pedance per  winding  of  the 
A-connected  generator. 

224.  Coil  Current  in  a 
A-Connected  Generator  for 
Balanced  Load.  — •  In  a  A- 
connected  generator  the 
current  7  taken  from  each 
terminal  of  the  generator 
Fig,  132.  must  be  equal  to  the  vector 

sum  of  the  currents  flowing  to  that  terminal  in  the  wind- 
ings of  the  generator.  Let  712,  723  and  73,  be  the  currents  in 

the  three  windings  in  the  directions  indicated  in  Fig.  133,  and 
as  before  let  701,  702  and  703  be  the  currents  taken  from  the  three 

terminals.  The  first  set  of  currents  are  called  the  "  A  "  or  coil 
currents  and  the  second  set  the  "  Y  "  or  line  currents.  When 
the  load  is  balanced,  the  three  Y  currents  are  equal  in  effective 
value  and  differ  in  phase  by  120°,  see  Article  223.  Hence, 
referring  all  the  currents  to  703  as  the  axis  of  reference,  and  as- 
suming sine-wave  currents  and  e.  m.  /.'s,  we  have  that 

•f  7  0     )  703  (a) 


Hence,  substituting  in  (a)  and  (6)  the  value  of  731  from  (c)  we  have 

(d) 


THREE-PHASE  ALTERNATING  CURRENTS 


405 


and 


Multiplying  (e)  by  2  and  adding  (d)  gives 

-3712=/\/3/03 
whence 


T    —  —     J       T 

1  12  -=     *  03 


Similarly  for  the  other  two  phases  ;  whence 


r 

~ 


(6) 


Hence  the  current  in  each  of  the  windings  of  a  A-connected  gener- 
ator is  equal  numerically  to  the  effective  value  of  the  Y  or  line 
current  divided  by  the  square  root  of  three,  and  lags  90°  behind 
the  current  in  the  opposite  branch  of  the  Y,  provided  the  system 
is  perfectly  balanced.  Compare  with  the  relation  between  the 
A  e.  m.  f.'s  and  the  Y  e.  m.  f.'s  given  by  equation  (3). 

The  Y  currents,  like  the  Y  e.  m.  f.'s,  may  be  represented  by 
three  equal  vectors  differing  in  phase 
by  120°.  The  sides  of  the  triangle 
formed  by  joining  the  ends  of  these 
vectors  are  then  equal  to  three  times  the 
currents,  but  their  directions,  taken 
positive  in  the  counter-clockwise  direc- 
tion,  give  the  proper  phase  relation  be- 
tween the  A  and  Y  currents,  see  Fig. 
133. 

225.  Coil  E.  M.  F.  and  Impedance 
of  the  Equivalent  Y-Connected  Gen- 
erator and  of  the  Equivalent  Y-Con- 
nected  Load. —  A  F-connected  and  a 

A-connected  generator  may  be  looked  upon  as  equivalent  to  each 
other  provided  the  voltage  between  terminals  is  the  same  in  each 
for  the  same  current  taken  from  the  terminals.  Let  EA,  7A  and 
ZA,  and  Ew  Iv  and  zv,  be  the  effective  e.m.f.  per  winding,  the  effective 


Fig.  133. 


406  ELECTRICAL  ENGINEERING 

current  per  winding  and  impedance  per  winding  for  a  A  and  Y 
connected  generator  respectively.  On  the  assumption  of  sine- 
wave  currents  and  e.  m.  f.'s,  the  voltage  between  terminals  and 
2  in  case  of  the  A-connected  generator  is  the  vector  difference 
7?i2— 2A  712,  and  the  voltage  between  terminals  in  the  case  of  the  Y- 
connected  generator  is,  when  referred  to  E12  as  the  line  of  reference, 
the  vector  difference  —  /  \/3  (Eos  —  zy  703),  provided  the  system  is 
balanced,  see  equations  (3).  Hence  in  order  that  these  two  ex- 
pressions be  equal  for  all  values  of  the  line  or  Y  current  we  must 
have 

#12  =  -/ V3  #03 

and  zJi2  =  -JV3zyIn 

But  from  equation  (6)  7J2  =  —  — -  703;  therefore 

«..  V    O     * 

*A=3*y  (7) 

Therefore,  in  order  that  the  voltage  between  terminals  be  the  same 
in  a  F-connected  as  in  a  A-connected  generator,  the  coil  e.  m.  /.  in 

the  F-connected  generator  must  be  equal  to   — =  times  the  coil 

V3 

e.  m.  /.  of  the  A-connected  generator  and  both  the  resistance  and 
the  reactance  of  each  winding  of  the  Y-connected  generator  must 
be  one-third  the  resistance  and  reactance  respectively  of  the 
A-connected  generator. 

By  a  similar  argument  it  can  be  shown  that  a  A-connected 
load  may  be  considered  equivalent  to  a  F-connected  load  pro- 
vided the  back  e.  m.  f.  in  each  winding  of  the  equivalent  Y  is 

taken  equal  to  — =  times  the  coil  e.  m.  /.  of  each  winding  of  the 

V3 

A-connected  load  and  both  the  resistance  and  reactance  of  each 
winding  of  the  equivalent  Y  are  taken  equal  to  one-third  the  re- 
sistance and  reactance  respectively  of  each  winding  of  the  A. 

226.  Reduction  of  all  Balanced  Three-Phase  Circuits  to  Equiv- 
alent Y's.  —  Any  problem  in  regard  to  a  balanced  three-phase 
circuit  may  therefore  be  solved  by  reducing  all  parts  of  the  circuit 
to  an  equivalent  Y  connection,  provided  the  currents  and  e.  m.  f.'s 
are  sine  waves.  The  transformations  are  made  as  follows : 

Any  A-connected  motor  or  generator  is  considered  as  equiv- 


THREE-PHASE  ALTERNATING  CURRENTS  407 


alent  to  a    F-connected  generator  or  motor  in  which  the  e.  m.  /. 

per  winding  Ey  is  equal  to  —  = 

V3 
of  the  A-connected  machine,  i.e., 


per  winding  Ey  is  equal  to  —  =  times  the  e.  m.  f.  per  winding 

V3 


(8a) 

and  the  impedance  per  winding  zy  of  the  equivalent  Y-connected 
machine  as  equal  to  the  impedance  per  winding  ZA  of  the  A-con- 
nected machine  divided  by  three,  i.e., 

z,=|  (86) 

Similarly,  any  A-connected  load  which  has  an  impedance 
per  winding  equal  to  fA  may  be  replaced  by  a  F-connected 
load  which  has  an  impedance  per  winding 


The  current  per  winding  7A  in  any  A-connected  generator  or 
load  is  equivalent  to  a  current  Iy  in  the  equivalent  F-connected 
generator  or  load,  where 

/,=V3/4  (8d) 

or  vice  versa,  when  the  current  in  the  equivalent  F  is  found  to 
be  Iy,  the  corresponding  current  per  winding  in  the  A  is 


> 

Similarly,  when  the  equivalent  F  voltage  is  found  to  be  Ey, 
the  corresponding  A  voltage  is 


Each  of  the  wires  (e.g.,  the  wire  3c  in  Fig.  131)  connecting  a 
generator  terminal  to  a  load  terminal  is  then  in  series  with  the  cor- 
responding phase  of  the  equivalent  F  generator  and  F  load. 

When  all  parts  of  the  circuit  have  thus  been  reduced  to  equiv- 
alent Y's,  each  of  the  three  phases  may  be  treated  as  a  single- 
phase  circuit,  each  circuit  considered  completed  by  a  wire  having 
zero  impedance  connecting  all  the  neutrals  together,  since  all 
the  neutrals  are  at  the  same  potential. 

227.  Power  in  a  Balanced  Three-Phase  System.  —  Consider 
first  a  F-connected  load,  and  let  the  potential  drop  per  winding 
be  Vy,  and  the  current  per  winding  Iy,  and  the  difference  in 


408  ELECTRICAL  ENGINEERING 

phase  between  current  and  p.d.  in  this  winding  be  9y,  then  the 
power  input  into  each  winding  is  Vyly  cos  8y,  and  therefore  the 
total  power  input  is 

P=3V»«w'0» 

Similarly,  in  the  case  of    a  A-connected  load,  for  which  TA  is 

the  potential  drop  per  winding,  7A  the  current  per  winding,  and 
#A  the  difference  in  phase  between  current  and  p.d.  in  this  winding, 
the  total  power  input  is 

P=37A/Acos0A 

In  the  case  of  a  generator  or  any  kind  of  a  load,  the  p.d.  that  is 
most  easily  measured  is  the  p.d.  between  terminals  or  the  p.d. 
between  the  mains  where  they  connect  with  the  terminals.  In 
case  of  a  A-connected  generator  or  load  this,  is  the  p.d.  per  wind- 
ing, that  is,  VA;  in  the  case  of  a  Y-connected  machine  this  is  the 
equivalent  A  p.d.  and  is  equal  to  the  p.d.  per  winding  Vy  mul- 
tiplied by  the  square  root  of  three,  that  is 


Since  FA  is  the  same  as  the  p.d.  between  wires,  VA  is  frequently 
called  the  line  voltage. 

The  current  that  is  usually  most  readily  measured  is  the 
current  in  each  main,  that  is,  the  current  leaving  each  terminal 
of  a  generator  or  entering  each  terminal  of  the  load.  In  the 
case  of  a  Y  connection  this  is  equal  to  the  coil  current  Iyi  and  in 
the  case  of  a  A-connected  machine  it  is  the  equivalent  Y  current 
and  therefore  is  equal  to  the  coil  current  7A  multiplied  by  the 
square  root  of  three.  That  is, 


Since  Iy  is  the  same  as  the  current  per  main,  Iy  is  frequently  called 
the  line  current.  Hence  in  expressing  the  power  input  into  a 
F-connected  load  it  is  more  convenient  to  substitute  for  Vy  its 
value  in  terms  of  the  line  voltage  VA,  which  gives 


\/3 

=  V3  7A  /„  cos  By  (9o) 

In  the  case  of  a  A-connected  load,  the  substitution  for  /A  its 
value  in  terms  of  the  line  current  Iy,  gives 


V3 


THREE-PHASE  ALTERNATING  CURRENTS  409 


A  /„  co«     A  (96) 

Equations  (9)  are  the  expressions  usually  employed  for  the 
three-phase  power,  and  are  usually  written  without  the  sub- 
scripts; namely,  the  total  three-phase  power  is  usually  written 

P  =VZ  V  I  cos  0 

It  should  be  clearly  borne  in  mind,  however,  that  the  6  in  this 
expression  is  not  the  phase  angle  between  the  voltage  V  and  the 
current  I,  but  is  the  phase  angle  between  the  voltage  to  neutral 
and  the  line  current,  which  in  turn  is  equal  to  the  phase  angle 
between  the  voltage  between  wires  and  the  A  current. 

228.  Example.  —  Energy  is  supplied  from  a  generating  station 
to  a  substation  50  miles  away  at  a  rate  of  20,000  kilowatts.  The 
system  is  a  balanced*  three-phase  system  and  operates  at  a  fre- 
quency of  25  cycles.  The  transmission  line  consists  of  three 
No.  0000  B.  &  S.  copper  wires  spaced  six  feet  between  centers. 
It  is  desired  to  find  (1)  what  will  be  the  voltage  between  wires 
at  the  generating  station  when  the  voltage  between  wires  at  the 
substation  is  60,000  volts,  and  the  power  factor  at  the  substation 
is  80  per  cent,  with  the  current  lagging,  (2)  how  much  power  is 
lost  in  the-  transmission  line,  and  (3)  what  is  the  power  factor 
at  the  generating  station.  The  electrostatic  capacity  of  the  line 
may  be  neglected. 

The  current  per  wire  is 

20,000,000 
/=  —  ^—  —  =241  amperes 

V3X  60,000X0.8 

The  voltage  from  line  to  neutral  at  the  substation  is 
F       60,000  _ 

\/3 

Taking  Iy  as  the  line  of  reference,  the  symbolic  notation  for  the 

voltage  to  neutral  is 

Vy  =34,600  (cos  6  +  j  sin  9) 

where  0  is  the  angle  by  which  the  voltage  leads  the  line  of 
reference  Iy  (see  Article  207).  But  cos  #  =  0.8,  and,  since  the 
current  is  lagging  the  angle  0  by  which  the  voltage  leads  the 
current  is  positive,  and  therefore  sin  #  =  4-0.6.  Therefore 

Vy  =34,600  (0.8  +  j  0.6)  =27,700  +  /  20,800 
The  resistance  per  mile  of  a  No.  0000  wire  is  0.258  ohms;  its 
inductance  per  mile  for  a  spacing  of  six  feet  is  1.93  millihenries, 
hence  the  reactance  per  mile  at  25  cycles  of  a  No.  0000  wire  is 


410  ELECTRICAL  ENGINEERING 

2  TT  X25X1.93X10'3=0.303  ohms.     Hence  the  total  impedance 
of  each  wire  is 

zy=50  X0.258+j50X0.303  =  12. 
Hence  the  voltage  to  neutral  at  the  generating  station  is 


=30,800  +  #4,500 

The  effective  value  of  which  is  39,300  volts,  and  therefore  the 
effective  voltage  between  wires  at  the  generating  station  is 

y/  =V3  X  39,300  =68,000 

The  power  lost  in  the  line  is  equal  to  3RIy2  where  R  is  the 
total  resistance  of  each  wire  and  Iy  the  line  current.  Hence  the 
power  lost  in  the  line  is 

3X  12.9  X  (241)2  watts  =2,250  kilowatts 

The  total  power  delivered  to  the  line  and  substation  is  then 
22,250  kilowatts.  Hence  the  power  factor  at  the  generating 
station  is 

22,250,000 


_ 
A/3  X  68,000X241 

229.  Rating  of  Three-Phase  Apparatus.  —  The  rated  voltage 
of  a  three-phase  machine  always  refers  to  the  volts  between  termi- 
nals or  the  A  voltage  EA,  the  power  rating  is  the  total  power 
for  all  three-phases,  and  the  power  factor  (usually  written  cos  0) 
is  the  ratio  of  the  total  power  P  to  the  square  root  of  three  times 
the  voltage  between  terminals  times  the  current  per  terminal 
Iy,  and  the  machine  is  assumed  to  carry  a  balanced  load.     The 
current  per  terminal  is  then 

/=      JL  _ 

V3  EA  cos  9 

which  is  also  the  current  per  winding  for  a  F-connected  machine. 
In  the  case  of  a  A-connected  machine  the  current  per  winding  is 

7  =-      P 

3  #A  cos  0 

230.  Measurement  of  Power  in  a  Three  -Phase  Circuit.  —  Two 
Wattmeter  Method.  —  To  measure  the  power  output  of  a  three- 
phase  generator  or  the  power  input  of  a  three-phase  load,  the 
most  obvious  method  is  to  measure  the  power  for  each  phase 
separately  by  connecting  the  current  coil  in  the  wattmeter  in 
series  with  the  winding  of  that  particular  phase  and  the  potential 
coil  of  the  wattmeter  across  the  terminals  of  that  winding.     The 


THREE-PHASE  ALTERNATING  CURRENTS 


411 


total  power  output  (or  input)  is  then  the  sum  of  the  outputs  (or 
inputs)  for  the  three  windings.  In  the  case  of  a  F-connected  load 
with  an  accessible  neutral  point  this  is  readily  accomplished, 
Fig.  134,  by  inserting  the  current  coil  in  series  with  the  line  and 
connecting  the  potential  coil  between  the.  terminal  of  load  and 
its  neutral  point.  (The  output  per  phase  of  a  F-connected 
generator  is  similarly  measured  by  connecting  the  current  coil 
in  series  with  that  phase  of  the  generator  and  the  potential  coil 
across  the  terminal  of  this  phase  and  the  neutral  point  of  the 
generator.)  In  case  the  load  is  balanced,  the  reading  of  the 
wattmeter  multiplied  by  three  gives  the  total  power.  When  the 
load  is  not  balanced  a  similar  measurement  must  be  made  for 
each  phase  and  the  three  readings  add- 
ed; the  three  readings  should  be  made  % 
simultaneously  if  the  load  is  varying. 

In  the  case  of  a  A-connected  gen-  Current  Coil 
erator  or  motor,  the  three  windings 
are  usually  connected  inside  the  ma- 
chine, and  it  is  therefore  not*  feasible 
to  connect  the  current  coil  in  series  with 
the  windings.  Again,  the  neutral  point 
of  a  F-connected  generator  or  mo.tor 
is  frequently  not  accessible.  The  fol- 
lowing method  for  measuring  the  power 
is  then  usually  employed,  and  gives  the 
true  power  whether  the  system  is  balanced 
or  not.  Two  wattmeters  A  and  B  are  connected  as  shown  in 
Fig.  135;  the  current  coil  of  A  is  connected  in  series  with  one 

ino 


Potential 
Coil 


Fig.  134. 


V32 


i23 


•  o 


Fig.  135. 


412  ELECTRICAL  ENGINEERING 

line  wire  a,  and  the  current  coil  of  B  is  connected  in  series  with 
a  second  line  wire  6.  The  potential  circuit  of  the  meter  A  is 
connected  across  from  a  to  the  third  line  wire  c,  and  the  potential 
circuit  of  meter  B  is  connected  across  from  the  line  b  to  the 
line  c.  Let  the  instantaneous  value  of  the  currents  and  p.d.'s 
be  represented  by  small  letters  with  subscripts  indicating  their 
directions.  The  instantaneous  torque  on  the  moving  element 
of  meter  A  is  then  proportional  to  v3l  i01  and  the  instantaneous 
torque  on  the  moving  element  of  meter  B  in  the  same  relative 
direction  is  v32  i^.  (That  is,  when  the  instantaneous  currents  and 
p.d.'s  are  actually  in  the  directions  indicated  by  the  arrows  on  the 
coils,  and  the  two  meters  are  exactly  alike,  these  are  the  values 
of  the  instantaneous  torques  in  each  meter  either  both  to  the 
left  or  both  to  the  right.) 

But  v32  =  —  v23 


Hence  the  total  torque  on  the  moving  elements  of  both  meters  is 

V3l  4l  +  ^32  ^02  =V3l    fel  -  t'l2)  —  ^23    O'l2  -  i2S) 

=  031  *31  +  ^23  *23  -  (081  +  ^23)    1*12 

But 

03i  +  023  =  -0i2 

(since  the  total  drop  of  electric  potential  at  any  instant  around 
a  closed  path  is  zero).  Hence  the  total  instantaneous  torque  on 
the  moving  elements  of  the  two  meters  is  proportional  to 

012*12  +  023^23  +  031*31 

which  is  equal  to  the  total  instantaneous  power. 

Since  the  reading  of  each  wattmeter  is  proportional  to  the 
average  torque  acting  on  its  moving  element,  it  follows  that  the 
algebraic  sum  (the  algebraic  sum,  since  the  average  values  of 
v31  iol  and  v23  iQ2  may  be  of  opposite  sign)  of  the  two  wattmeter 
readings  gives  the  true  average  power,  independent  of  whether 
the  load  is  balanced  or  not  and  also  independent  of  the  wave 
shape  of  current  and  p.d.;  provided  the  wave  shape  of  the  current 
in  the  potential  circuit  is  the  same  as  the  wave  shape  of  the  p.d. 
across  it.  This  provision  is  only  approximately  fulfilled  when 
the  potential  circuit  of  the  meter  contains  iron. 

Since  the  average  value  of  the  instantaneous  products  v31  iQl  and 
v32  i02  may  be  either  positive  or  negative,  the  deflection  of  the  watt- 
meter needle  may  be  either  to  the  right  or  to  the  left  of  the  zero. 


THREE-PHASE  ALTERNATING  CURRENTS  413 

Wattmeters,  however,  are  usually  designed  with  a  scale  reading 
only  to  the  right ;  consequently  when  a  positive  value  of  the  aver- 
age of  the  product  vi  corresponds  to  a  deflection  to  the  right,  then 
the  needle  goes  off  the  scale  to  the  left  when  the  value  of  this 
product  is  negative.  However,  by  reversing  the  leads  connecting 
the  current  coil  in  the  line,  the  direction  of  the  current  in  the  cur- 
rent coil  is  reversed  with  respect  to  the  current  in  the  potential 
coil,  and  the  needle  will  then  deflect  to  the  right.  (Note  that  the 
terminal  of  the  potential  circuit  connected  directly  to  the  sus- 
pended coil  of  the  meter  should  always  be  connected  to  the  line 
wire  in  which  the  current  coil  of  the  meter  is  connected ;  otherwise, 
since  the  impedance  of  this  coil  is  only  a  fraction  of  the  total 
impedance  of  the  potential  circuit,  practically  the  full  voltage 
across  the  potential  circuit  will  also  exist  between  the  potential 
coil  and  the  current  coil  of  the  meter,  and  may  cause  the  insulation 
between  the  two  to  break  down.) 

To  measure,  then,  the  total  three-phase  power  by  this  two 
wattmeter  method,  it  is  necessary  to  connect  the  current  coils  of 
the  two  wattmeters  in  the  lines  a  and  b  in  such  a  manner  that  the 
needles  of  both  meters  deflect  to  the  right.  It  is  then  necessary 
to  determine  whether  the  two  readings  shall  be  added  or  sub- 
tracted. This  is  determined  from  the  fact  that  if  the  average 
value  of  the  two  products  v31  iol  and  vS2  i02  are  both  of  the  same 
sign,  then  both  meters  will  also  read  to  the  right  if  they  are  inter- 
changed, and  the  connection  to  the  middle  wire  c  is  kept  unaltered. 
If,  however,  the  average  values  of  these  two  products  are  of  the 
opposite  sign,  then  when  the  two  meters  are  interchanged  and  the 
connection  to  c  is  kept  unaltered,  the  deflection  of  each  instrument 
will  reverse.  Hence,  the  general  rule,  subtract  the  two  readings  if 
on  substituting  one  meter  for  the  other  (the  connection  to  the  common 
wire  c  being  kept  unaltered],  the  deflection  reverses;  otherwise  the  two 
readings  are  to  be  added.  • 

231.  Two  Wattmeter  Method  Applied  to  a  Balanced  Three- 
Phase  System.  —  In  case  the  load  is  balanced  and  the  currents 
and  p.d.'s  are  both  harmonic  functions  or  sine  waves,  the  phase 
relations  of  the  currents  and  potential  drops  in  the  two  watt- 
meters can  be  readily  deduced.  Let  6  be  the  angle  by  which  the 
line  current  Iy  lags  behind  the  potential  drop  Vy  (i.e.,  cos  6  is 
the  power  factor  of  the  load),  then  the  vectors  representing  the 
various  currents  and  p.d.'s  are  as  shown  in  Fig.  136. 


414 


ELECTRICAL  ENGINEERING 


The  average  value  of  the  instantaneous  product  v3l  iol  is  then 
equal  to  the  product  of  the  lengths  of  the  two  vectors  representing 
v3l  and  v  by  the  cosine  of  the  angle  between  them,  that  is 

Pl  =  average  (vB1  iol)  =  7A  Iy  cos  (30°  +  0) 

Similarly,  the  average  value  of  the  instantaneous  product  v32  ioz 
is  equal  to  the  product  of  the  lengths  of  the  two  vectors  repre- 
senting v32  and  in2  by  the  cosine  of  the  angle  between  them,  that  is 

P2  =  average  (v32  i02)  =  VJy  cos  (30°-  ff) 

Hence  when  6  lies  between  -  60°  and  +  60°  both  Pl  and  P2  are 
positive  and  therefore  their  sum  gives  the  true  three-phase  power ; 
when  6  is  less  than  —  60°  or  greater  than  +  60°,  Pj  and  P2  are 
of  opposite  sign,  and  hence  their  difference  gives  the  true  three- 
phase  power.  But  cos  60°  =i;  hence  when  the  power  factor  of 


Fig.  136. 

the  load  is  greater  than  50  per  cent  either  leading  or  lagging,  the 
sum  of  the  wattmeter  readings  gives  the  true  power;  when  the 
power  factor  of  the  load  is  less  than  50  per  cent  the  difference  of 
the  wattmeter  readings  gives  the  true  power.  * 

In  general,  the  power  factor  of  the  load  is  not  known.     How- 

*Note  that  in  the  above  discussion  a  negative  value  of  0  means  a  leading 
current. 


THREE-PHASE  ALTERNATING  CURRENTS  415 

ever,  in  the  case  of  a  balanced  load  the  following  simple  test  may 
be  used  to  determine  whether  the  wattmeter  readings  are  to  be 
added  or  subtracted.  Leaving  all  other  corrections  unchanged, 
transfer  the  connection  of  the  potential  coil  of  meter  A,  say,  from 
the  common  wire  c  to  the  wire  b.  Then  this  meter  will  read  the 
average  value  of  the  product  v2l  iol,  and  therefore  its  reading  will  be 

P/  =  VAIycos  (30°  -0) 

Hence  when  -60  <  9  <  +60,  Pl  and  P/  will  both  be  of  the  same 
sign;  when  0  <  -60  or  0  >  +  60,  P,  and  P/  will  be  of  the  op- 
posite sign.  Consequently,  if  when  this  change  in  connection  is 
made,  the  deflection  of  meter  A  remains  in  the  same  direction, 
the  sum  of  the  original  readings  of  A  and  B  gives  the  true  power  ; 
if  the  deflection  of  A  reverses,  then  the  difference  of  the  original 
readings  of  A  and  B  gives  the  true  power.  An  inspection  of  the 
vector  diagram  will  show  that  this  same  rule  applies  to  meter 
B  when  the  connection  of  the  potential  coil  of  B  to.  the  common 
wire  c  is  transferred  to  the  wire  a,  all  other  connections  remaining 
unaltered.  Hence  the  general  rule  for  a  balanced  load;  connect 
the  two  meters  in  circuit  in  such  a  manner  that  both  deflect  to 
the  right  and  take  the  two  readings  Pl  and  P2.  Then,  keeping 
all  other  connections  unaltered,  transfer  the  connection  of  the 
potential  coil  of  one  meter  from  the  common  wire  to  the  wire  in 
which  the  current  coil  of  the  second  meter  is  connected;  if  the 
deflection  of  the  former  meter  remains  in  the  same  direction  add 
the  original  readings  of  the  two  meters  ;  if  the  deflection  of  this 
meter  reverses,  subtract  the  original  readings  of  the  meters. 

In  the  case  of  a  balanced  load  the  power-factor  angle  9  may 
be  expressed  directly  in  terms  of  the  wattmeter  readings.  For, 
taking  the  sum  and  difference  respectively  of  P2  and  Pl,  we  get 


P2  +  pi  =  vjy  [cos  (30°-  0)  +  cos  (30°+  0)]  =^/3VJy  cos  9 
P2  -  P,  =  VJy  [cos  (30°  -  0)  -  cos  (30°  +  9)]  =  VJV  sin  9 
whence  taking  the  ratio  of  these  two  expressions,  we  get 


416  ELECTRICAL  ENGINEERING 

SUMMARY  OF  IMPORTANT  DEFINITIONS  AND 
RELATIONS 

1.  The  distance  between  the  centers  of  successive  magnetic 
poles  of  like  sign  in  any  electric  machine  is  said  to  correspond  to 
360  electrical  degrees. 

2.  A  two-phase  generator  has  two  independent  windings  on  its 
armature  displaced  90  electrical  with  reference  to  each  other;   a 
three-phase  generator  has  three  independent  windings  on  its  arma- 
ture, successive  windings  being  spaced  120  electrical  degrees  with 
reference  to  each  other. 

3.  A  A-connected  generator  has  its  three  windings  connected 
in  series  and  terminals  brought  out  at  the  three  junction  points  ; 
a  F-connected  generator  has  one  end  of  each  of  its  windings  con- 
nected to  a  common  junction  point  called  the  neutral  and  the 
other  ends  brought  out  to  terminals. 

4.  The  sum  of  the  e.  m.  f.'s  induced  in  the  three  windings  of  an 
alternator  when  these  e.  m.  f.'s  are  sine  waves  is  zero. 

5.  The  equivalent  A  e.  m.  f.  of  a    Y-connected  alternator  is 
numerically  equal  to  the  square  root  of  three  times  the  effective 
value  of  the  Y  e.  m.  /.  and  lags  90°  behind  the  e.  m.  f.  in  the  op- 
posite branch  of  the   Y;   that  is, 


provided  the  e.  m.  f.'s  are  sine  waves. 

6.  The  equivalent  Y  e.  m.  f.  of   a  A-connected  alternator  is 
numerically  equal  to  the  effective  value  of  the  A  e.  m.  f.  divided 
by  the  square  root  of  three  and  leads  by  90°  the  e.  m.  f.  in  the 
opposite  side  of  the  delta  ;  that  is, 

E«=& 
\/3 

provided  the  e.  m.  f.'s  are  sine  waves. 

7.  A  three-phase  system  is  said  to  be  balanced  when  the  cur- 
rents and  e.  m.  f.'s  respectively  in  the  three  phases  are  equal  in 
effective  value  and  differ  in  phase  by  120°. 

8.  The  equivalent  A  current  of  a  F-connected  alternator  (or 
load)  is  numerically  equal  to  the  effective  value  of  the  Y  current 
divided  by  the  square  root  of  three  and  lags  90°  behind  the  cur- 
rent in  the  opposite  branch  of  the  Y  ;  that  is 


- 
\/3 


THREE-PHASE  ALTERNATING  CURRENTS  417 

provided  the  currents  and  e.  m.  f.'s  are  sine  waves  and  the  system 
is  balanced. 

9.  The  equivalent  Y  current  of  a  A-connected  alternator  (or 
load)  is  numerically  equal  to  the  square  root  of  three  times  the 
effective  value  of  the  A  current  and  leads  by  90°  the  current  in 
the  opposite  side  of  the  A;  that  is, 


provided  the  currents  and  e.  m.  f.'s  are  sine  waves  and  the  system 
is  balanced. 

10.  The  equivalent  A  impedance  of  a   Y-connected  winding  is 
equal  to  three  times  the  Y  impedance,  that  is, 

11.  The  equivalent  Y  impedance  of  a  A-connected  winding  is 
equal  to  the  A  impedance  divided  by  three,  that  is, 

'     ~3  '  A 

provided  the  currents  and  e.  m.  f.'s  are  sine  waves  and  the  system 
is  balanced. 

12.  When  all  parts  of  a  balanced  three-phase  system  have  been 
reduced  to  equivalent    Y's,  each  of  the  three  phases  may  be 
treated  as  a  single  phase  circuit,  each  such  circuit  being  considered 
completed  by  a  wire  of  zero  impedance  connecting  all  the  neutrals, 
provided  the  currents  and  e.  m.  f.'s  are  sine  waves. 

13.  The  total  power  input  into  a  three-phase  load  is 

P=V3VIcos0 

where  V  is  the  p.d.  between  terminals  (the  A  p.d.),  I  is  the  line 
current  (the  Y  current)  and  6  is  the  power-factor  angle  (the 
difference  in  phase  between  the  Y  p.d.  and  Y  current  or  between 
the  A  p.d.  and  A  current).  This  relation  holds  only  when  the 
currents  and  p.d.'s  are  sine  waves. 

14.  When  two  wattmeters,  connected  in  a  three-phase  circuit 
as  shown  in  Fig.  135,  read  Pl  and  P2  watts  respectively,  the  total 
power  is 

P=Pl±P2 

To  determine  which  sign  to  use,  substitute  one  meter  for  the  other, 
keeping  the  connection  to  the  common  wire  unchanged;  if  the 
deflection  of  this  meter  reverses  take  the  difference,  if  the  deflec- 
tion does  not  reverse  take  the  sum. 


418  ELECTRICAL  ENGINEERING 

When  the  system   is  balanced  and  the  currents  and  e.  m.  f.'s 
are  sine  waves,  the  power  factor  of  the  load  is  cos  6  where 

p  _  p 
2      * 


PROBLEMS 

1.  The  ends  of  the  three  coils  of  a  three-phase  alternator  are 
brought  out  to  six  slip  rings.     When  the  three  coils  on  this  alterna- 
tor are  connected  in   Y  and  the  machine  is  operated  at  its  rated 
speed  and  field  excitation,  the  rated  output  of  the  machine  is 
30  kw.  at  1000  volts.     If  the  speed  and  field  excitation  are  kept 
constant,  what  would  be  (1)  the  rated  output  of  the  machine,  and 
(2)  the  rated  voltage  when  the  three  coils  are  connected  in  A? 

Ans.:  (1)  30  kw. ;  (2)  577  volts. 

2.  A  25-cycle,  three-phase,  A-connected  generator  develops  an 
e.  m.  f.  per  coil  of  1000  volts  effective  value.     This  e.  m.  /.  con- 
tains the  third  harmonic  and  the  effective  value  of  this  harmonic 
is  10  per  cent  of  the  effective  value  of  the  resultant  e.  m.  f.     The 
resistance  per  coil  of  the  generator  is  0.1  ohm  and  the  reactance 
per  coil  at  25  cycles  is  0.3  ohm.     (1)  What  will  be  the  value  of  the 
current  per  coil  when  the  generator  is  supplying  no  external  load? 
(2)  Will  this  current  be  a  sine-wave  current,  and  if  so,  what  will 
be  its  frequency? 

Ans.:  (1)  110.4  amperes  ;  (2)  sine-wave  current  with  frequency 
of  75  cycles. 

3.  A  500-volt,  25-cycle,  100-kilovolt-ampere,  three-phase,  A- 
connected  alternator,,  while  operating  at  full  load  on  a  circuit 
of  85%  power  factor,  has  one  of  its  coils  burnt  out.     With  this 
coil  open-circuited,  (1)  what  is  the  maximum  balanced  load  at 
85%  power  factor  which  the  machine  can  supply  continuously? 
(2)  What  proportion  of  this  load  is  supplied  by  each  of  the  two 
remaining  coils? 

Ans.:  (1)  49.1  kw. ;  (2)  32.1%  and  67.9%. 

4.  The  armature  of  a  230-volt,  two-phase,  50-kilovolt-ampere 
alternator  is  re-wound  so  that  the  end  of  one  of  the  coils,  A,  is 
connected  to  the  middle  of  the  other  coil,  B.     If  13.4%  of  the 
turns  on  coil  B  are  removed,  what  will  be  the  rating  of  the  re- 
sultant alternator? 

Ans.:  230  volts,  3-phase,  43.3  kilo  volt-amperes. 


THREE-PHASE  ALTERNATING  CURRENTS  419 

5.  A  coil  of  high  impedance  is  connected  between  two  slip-rings 
of  a  F-connected  200-volt  alternator.   Find(l)  the  potential  differ- 
ence between  the  middle  point  of  the  impedence  coil  and  the 
neutral  point  of  the  alternator,  and  (2)  the  potential  difference 
between    the    middle    point    and    the    third    terminal    of    the 
alternator. 

Ans.:  (1)  57.8  volts;  (2)  173.2  volts. 

6.  The  potential  difference  between  each  of  the  line  wires  A, 
B  and  C  and  the  neutral  0  of  a  F-connected  load  on  a  three-phase 
system  is  100  volts  and  these  respective   F  voltages  differ  in  phase 
by  120°.      The  impedances   of  the   three   branches   of  the  load 
are  respectively  ZAO  =  W  +  jQ,  ZBO=Q+jlQ  and  Zco=0-/10. 
(1)  What  will  an  ammeter  connected  between  the  neutral  of  the 
load  and  the  neutral  of  the  generator  indicate?    (2)  If  the  cur- 
rent coils   of  two  wattmeters  are  connected  in  lines   A  and  B 
respectively  and  the  potential  coils  are  connected  from  A  to  C 
and  fro:n   B  to  C  respectively,  what  will  each  wattmeter  read? 
(3)  What  is  the  total  power  supplied  to  the  load? 

Ans.:  (1)   7.32  amperes;   (2)  1500  watts  and  866  watts;   (3) 
1000  watts. 

7.  A  three-phase  F-connected  generator  supplies  power  to  a  bal- 
anced load  at  a  line  voltage  of  220  volts.     The  power  is  measured 
by  the  two  wattmeter  method  and  each  wattmeter  reads  10  kw. 
If  the  resistance  and  reactance  of  each  phase  of  the  armature  is 
0.1  ohm  and  0.3  ohm  respectively,  find  the  induced  e.  m.  f.  per 
phase  of  the  alternator. 

Ans.:  133.2  volts. 

8.  In  series  with  each   terminal  of   a  balanced   three-phase 
load  is  connected  a  non-inductive  resistance  of  0.5  ohm.     The  p.d. 
between  the  terminals  of  the  load  is  200  volts,  the  p.d.  between 
line  wires  just  outside  the  resistances  is  300  volts,  and  the  p.d. 
across  the  terminals  of  each  resistance  100  volts.    What  is  (1)  the 
total  power  supplied  to  the  load,  and  (2)  its  power  factor? 

Ans.:  (1)  20  kw. ;  (2)  28.9%. 

9.  Each  phase  of  a  balanced  three-phase,  delta-connected  load 
is  formed  by  an  impedance  coil  in  series  with  a  non-inductive 
resistance  of  15  ohms.     The  voltage  across  the  impedance  coil  in 
each  phase  is  80,  the  voltage  across  the  non-inductive  resistance  in 
each  phase  is  150,  the  resultant  voltage  across  the  impedance  coil 
and  the  non-inductive  resistance  is   200.     What  would  be  the 


420  ELECTRICAL  ENGINEERING 

readings  of  the  two  wattmeters  when  connected  to  this  load  to 
measure  the  power  by  the  two-wattmeter  method? 
Am.:  3420  watts  and  2190  watts. 

10.  The  two  wattmeters  connected  to  a  balanced  three-phase 
load  read  5240  watts  and  2380  watts  respectively.     (1)  What  is  the 
difference  in  phase  between  the  current  in  the  current  coil  of  the 
first  wattmeter  and  the  potential  drop  through  the  potential  coil 
of  the  second  wattmeter?    (2)  What  is  the  power  factor  of  the 
load? 

.    Ans.:  (1)  123  degrees  or  57  degrees ;  (2)  83.9%. 

11.  The  current  coil  of  a  single  wattmeter  is  connected  in  one 
of  the  line  wires  of  a  balanced  three-phase  load  which  takes  30 
kilowatts  at  a  lagging  power  factor  of  80%.      Call  this  wire  A 
and  the  other  two  line  wires  B  and  C  respectively.     What  would 
the  wattmeter  read  when  its  potential  coil  is  connected  (1)  between 
A  and  B,  (2)  between  A  and  C,  and  (3)  between  B  and  C? 

Ans.:  (1)  21.5  kilowatts;  (2)  8.5  kilowatts;  and  (3)  13.0  kilo- 
watts. 

12.  20,000  kilowatts  is  supplied  from  a  generating  station  to  a 
substation  50  miles  distant  over  a  three-phase  line  formed  of 
three  No.  0000  B.  &  S.  copper  wires  spaced  6  feet  between  cen- 
ters.    The  resistance  of  each  wire  per  mile  is  0.258  ohms  and  the 
inductance  per  mile  of  each  wire  is  1.93  millihenries.     A  voltage 
of  60,000  is  maintained  between  wires  at  the  substation  and  the 
power  factor  of  the  substation  is  80%  lagging  current.      The 
system  is  balanced     Calculate  (1)  the  voltage  between  wires  at 
the  generating  station,  (2)  the  total  number  of  kilowatts  lost  in 
the  transmission  line  and  (3)  the  power  factor  at  the  generating 
station. 

Ans.:  (1)  68,200  volts;  (2)  2250  kilowatts;  (3)  78.3%. 

13.  Energy  is  supplied  from  a  three-phase  Y-connected  gen- 
erator over  a  three-phase  line  to  a  balanced  three-phase  load 
formed  by  two  motors  connected  in  parallel.     The  motors  take 
respectively  50  kilowatts  at  80%  power  factor  lagging  and  100 
kilowatts  at  90%  power  factor  leading.     The  potential  difference 
between  the  terminals  of  the  motors  is  500  volts  and  the  resistance 
and  reactance  of  each  line  wire  are  respectively  0.1  ohm  and  0.2 
ohm.     (1)  What  is  the  symbolic  expression  for  the  total  current 
per  wire  taken  by  the  load  referred  to  the  potential  drop  be- 
tween the  terminal  of  one  terminal  of  the  load  and  the  neutral? 


THREE-PHASE  ALTERNATING  CURRENTS  421 

(2)  What  is  the  numerical  value  of  the  potential  difference  be- 
tween the  generator  terminal  and  the  neutral? 

Ans.:  (1)  173.3  +  y0.3;  (2)  308  volts. 

14.  A  three-phase  Y-connected  generator  delivers  power  to  a 
balanced  delta-connected  load  over  a  transmission  line.  The  resist- 
ance of  each  line  wire  is  0.3  ohm  and  the  reactance  of  each  line 
wire  is  0.5  ohm.  The  potential  differences  at  the  terminals  of  the 
generator  and  the  load  are  respectively  500  volts  and  400  volts. 
The  power  factor  of  the  load  is  80%  lagging.  Calculate  (1)  the 
line  current,  (2)  the  power  delivered  to  the  load,  (3)  the  power 
output  of  the  generator,  (4)  the  power  factor  of  the  generator  and 
(5)  the  efficiency  of  the  transmission  line  at  this  load. 

Ans.:  (1)  104.7  amperes ;  (2)  58.0  kilowatts ;  (3)  67.9  kilowatts  : 
(4)  78.4%;  (5)  85.4%. 


APPENDIX   A 


THE  following  abbreviations  are  recommended  by  the  American  Institute 
of  Electrical  Engineers  and  are  used  in  all  its  publications.  In  general  these 
abbreviations  should  be  used  only  when  expressing  definite  numerical  values. 


NAME 
Alternating  current 

Amperes 

Brake  horse  power 

Boiler  horse  power 

British  thermal  units 

Candle-power 

Centigrade 

Centimetres 

Circular  mils 

Counter  electromotive  force 

Cubic 

Diameter 

Direct  current 

Electric  horse  power 

Electromotive  force 

Fahrenheit 

Feet 

Foot-pounds 

Gallons 

Grains 

Grammes 

Gramme-calories 

High-pressure  cylinder 

Hours 

Inches 

Indicated  horse  power 

Kilogrammes 

Kilogramme-metres 

Kilogramme-calories 

Kilometres 

Kilowatts 

Kilowatt-hours 

Magnetomotive  force 

Mean  effective  pressure 

Miles 


INSTITUTE  STYLE 
spell  out,  or  a-c.  when  used  as  compound 

adjective 
spell  out 
b.h.p. 
Boiler  h.p. 
B.t.u. 
c-p. 
cent. 
cm. 

cir.  mils 
counter  e.m.f. 
cu. 

spell  out 
spell  out,  or  d-c.  when  used  as  compound 

adjective 
e.h.p. 
e.m.f. 
fahr. 
ft. 

ft-lb. 
gal. 
gr. 
g- 

g-cal. 
spell  out 
hr. 
in. 
i.h.p. 
kg- 
kg-m. 
kg-cal. 
km. 
kw. 
kw-hr. 
m.m.f. 
spell  out 
spell  out 


422 


APPENDIX  A 


423 


NAME 

Miles  per  hour  per  second 
Millimetres 
Milligrammes 
Minutes 
Metres 

Metre-kilogrammes 
Microfarad 
Ohms 
Per 

Percentage 
Pounds 
Power-factor 
Revolutions  per  minute 
Seconds 
Square 

Square- root-of-mean-square 
Ton-mile 
Tons 
Volts 

Volt- amperes 
Kilovolts 
Kilovolt-amperes 
Watts 
Watt-hours 

Watts  per  candle-power 
Yards 


INSTITUTE  STYLE 
Miles  per  hr.  per  sec. 
mm. 
mg. 
min. 
m. 

m-kg. 
spell  out 
spell  out 
spell  out 

per  cent,  or  %  in  tabular  matter  only 
Ib. 

spell  out 

rev.  per  min.,  or  r.p.m. 
sec. 
sq. 

effective,  or  r.m.s. 
spell  out 
spell  out 
spell  out 
spell  out 
kv. 
kv-a. 
spell  out 
watt-hr. 
watts  per  c-p. 


yd. 

1.  Use  lower  case  characters  for  abbreviations.     An  exception  to  this 
rule  may  be  made  in  the  case  of  words  spelled  normally  with  a  capital. 
Example:  "B.t.u."  and  not  "b.t.u."  nor  "B.T.U."    (British  thermal   unit). 
"U.S.gal."  (United  States  gallon);  "B.  &  S.  gauge"  (Brown  &  Sharpe  gauge). 

2.  Use  all  abbreviations  in  the  singular.     Example:   "17  Ib."  and  not 
"17  Ibs."  (17  pounds).     "14  in."  not  "14  ins."  (14  inches). 

3.  Use  a  hyphen  to  connect  abbreviations  in  cases  where  the  words  would 
take  a  hyphen  if  written  out  in  full.     When  a  hyphen  is  used,  omit  the 
period  immediately  preceding  the  hyphen.     Example:  "3  kw-hr."  and  not 
"3  kw.-hr."  (3  kilowatt-hours). 

4.  Use  a  period  after  each  abbreviation.     In  a  compound  abbreviation, 
do  not  use  a  space  after  the  period.     Example:  "i.h.p."   and  not  "i.  h.  p." 
(indicated  horse  power). 

5.  Never  use  "P."  for  "per"   but  spell  out  the  word.     Example:  "100 
ffr-lb.  per  ton"  (100  foot-pounds  per  ton);  "60  miles  per  hr."  (60  miles  per 
hour). 

6.  Use  "Fig."  not  "Figure."     Example:  "Fig.  3"   and  not  "Figure  3." 

7.  In  all  decimal  numbers  having  no  units,  a  cipher  should  be  placed 
before  the  decimal  point.     Example:  "0.32  Ib."  not  " .32  Ib." 

8.  Use  the  word  "by"  instead  of  "x"  in  giving  dimensions.      Example: 
"8  by  12  in."  not  "8  x  12  in." 


424  APPENDIX  A 

9.  Never  use  the  characters  (')  or  (")  to  indicate  either  feet  and  inches, 
or  minutes  and  seconds  as  period  of  time. 

10.  Use  capitals  sparingly;  when  used  as  units,  do  not  capitalize  volt, 
ampere,  watt,  farad,  henry,  ohm,  coulomb,  etc. 

11.  Do  not  use  the  expression  "rotary"  or  "rotary  converter";  use  "con- 
verter" or  " synchronous  converter." 

12.  Do  not  use  a  descriptive  adjective  as  a  synonym  for  the  noun  described. 
Example:  a  "spare  transformer,"  not  a  "spare";  a  "portable  instrument," 
not  a  "portable";  "automatic  apparatus,"    not  "automatics";  a  "short  cir- 
cuit," not  a  "short." 

13.  Do  not  use  the  expressions  a.c.  current  or  d.c.  current;  a.c.  voltage 
and  drc.  voltage.     Their  equivalents,  "alternating-current  current,"  "direct- 
current   current,"    "alternating-current    voltage,"    "direct-current   voltage" 
are  absurdities. 

14.  Do  not  use  the  expressions,     raising  transformer,"  "lowering  trans- 
former";   these    expressions    are    ambiguous.      Use   "step-up    transformer," 
"step-down  transformer." 

15.  Do  not  use  the  words  "primary"  and  "secondary"  in  connection  with 
transformer  windings.     Use  instead  " high-tension  "  and  "low- tension.' ' 


NOTATION. 

The  following  notation  forms    1324  of  Section  V  of  the  Standardization 
Rules  of  the  Institute. 

E,  e,  voltage,  e.m.f.,  potential  difference 

I,  i,  current  Z,  z,  impedance 

P,  power  L,  I,  inductance 

<£,  magnetic  flux  C,  c,  capacity 

ft,  B,  magnetic  density  Y,  y,  admittance 

R,  r,  resistance  6,  susceptance 

x,  reactance  G,  g,  conductance 


APPENDIX    B 

IN  this  country  wires  for  electrical  purposes,  when  less  than  half  an  inch 
in  diameter,  are  nearly  always  specified  in  terms  of  a  wire  gauge  introduced  by 
the  Brown  and  Sharpe  Manufacturing  Co.  This  gauge,  called  briefly  the  B.  & 
S.  gauge,*  is  such  that  successive  sizes  differ  in  diameter  by  a  constant  per- 
centage. A  solid  wire  460  mils  in  diameter  is  called  a  No.  0000  wire  and  a  wire 
5  mils  in  diameter  is  called  a  No.  36  wire.  The  next  smaller  size  to  a  No.  0000 
wire  is  No.  000,  the  next  smaller  size  No.  00,  the  next  No.  0,  the  next  No.  1 
and  so  on  up  to  No.  36.  The  ratio  of  the  diameters  of  No.  0000  and  No.  36 
is  4|p-— 92,  and  the  ratio  of  the  diameters  of  successive  sizes  is  constant;  this 
constant  is  therefore  equal  to  the  39th  root  of  92.  The  39th  root  of  92  is 
approximately  equal  to  the  sixth  root  qf  2;  hence  the  following  approximate 
relations  (since  the  cross  section  varies  as  the  square  of  the  diameter  and  the 
cube  root  of  2  is  approximately  1.26) : 

1 .  The  ratio  of  the  cross  sections  of  wires  of  successive  sizes  on  the  B.  & 
S.  gauge  is  equal  to  1 .26,  the  larger  number  on  the  gauge  corresponding  to 
the  smaller  cross  section.     This  same  relation  holds  for  the  weights  of  succes- 
sive sizes  for  a  given  length. 

2.  The  ratio  of  the  resistances  of  wires  of  successive  sizes  on  the  B.  &  S. 
gauge  is  equal  to  1.26,  the  larger  number  on  the  gauge  corresponding  to  the 
larger  resistance. 

3.  An  increase  of  3  in  the  gauge  number  halves  the  cross  section  and 
weight  and  doubles  the  resistance. 

4.  An  increase  of  10  in  the  gauge  number  decreases  the  cross  section  and 
weight  to  one-tenth  their  original  values  and  increases  the  resistance  to  10 
times  its  original  value. 

The  cross  section  of  a  No.  10  wire  is  approximately  10,000  circular  mils. 
The  resistance  of  this  size  copper  wire  is  approximately  1  ohm  per  1000  feet 
at  20°  cent,  and  its  weight  approximately  31.5  pounds  per  1000  feet.  From 
the  above  relations  the  resistance,  cross  section  and  weight  of  any  size  of  wire 
may  be  calculated  approximately  with  but  little  effort.  The  resistance  of  a 
No.  10  aluminum  wire  is  approximately  1.6  ohms  per  1000  feet  at  20°  cent, 
and  its  weight  approximately  15  pounds  per  1000  feet. 

The  above  relations  are  for  solid  wire.  Stranded  wire  is  numbered  on  the 
B.  &.  S  gauge  in  accordance  with  its  cross  section,  not  its  diameter.  The 
diameter  of  a  concentric  strand  is  approximately  15%  greater  than  that  of  a 
solid  wire  of  the  same  number;  its  weight  and  resistance  are  from  1  to  2  per 
cent  greater,  depending  upon  the  number  of  twists  per  unit  length  and  the 
number  of  wires  in  the  strand. 

Below  are  given  the  exact  relations  between  gauge  numbers,  diameter, 
cross  section,  weight  and  resistance  for  copper  wire  of  100%  conductivity 
Matthiessen's  Standard.  For  aluminum  wire  of  62%  conductivity  multiply 
the  weights  by  0.47  and  the  resistances  by  1.613. 

*  This  gauge  is  also  called  the  American  Wire  Gauge,  abbreviated  A.  W.  G. 

425 


426  APPENDIX  B 

Solid  Copper  Wire  —  100%  Matthiessen's  Standard 


Diam. 

Mils 

Weight,  Pounds 

Resistance,  20°  C. 
68°  F. 

No. 

. 

B.&  S. 

Area 
Cir.  Mils 

TTpof 

Bare 

1000' 

Mile 

j.  "Cu 

per  Ib. 

1000' 

Mile 

0000 

460 

211,600 

640.5 

3,381 

1.561 

0.04913 

0.2594 

000 

409.6 

167,800 

508 

2,682 

1.969 

0.06195 

0.3271 

00 

364.8 

133,100 

402.8 

2,127 

2.482 

0.07811 

0.4124 

0 

324.9 

105,500 

319.5 

1,687 

3.130 

0.09850 

0.5210 

1 

289.3 

83,690 

253.3 

1,337 

3.947 

0.1242 

0.6557 

2 

257.6 

66,370 

200.9 

1,062 

4.977 

0.1566 

0.8270 

3 

229.4 

52,630 

159.3 

841.1 

6.276 

0.1975 

1.043 

4 

204.3 

41,740 

126.4 

667.4 

7.914 

0.2490 

1.315 

5 

181.9 

33,100 

100.2 

529.0 

9.980 

0.3141 

1.658 

6 

162.0 

26,250 

79.46 

419.5 

12.580 

0.3960 

2.091 

7 

144.3 

20,820 

63.02 

332.7 

15.87 

0.4993 

2.636 

8 

128.5 

16,510 

49.98 

263.9 

20.01 

0.6296 

3.324 

9 

114.4 

13,090 

39.63 

209.2 

25.23 

0.7940 

4.192 

10 

101.9 

10,380 

31.43 

166.0 

31.82 

1.001 

5.286 

11 

90.74 

8,234 

24.93 

131.6 

40.12 

1.262 

6.664 

12 

80.81 

6,530 

19.77 

104.4 

50.59 

1.592 

8.407 

13 

71.96 

5,178 

15.68 

82.79 

63.79 

2.008 

10.60 

14 

64.08 

4,107 

12.43 

65.63 

80.44 

2.531 

13.36 

15 

57.07 

3,257 

9.858 

52.05 

101.4 

3.192 

16.85 

16 

50.82 

2,583 

7.818 

41.28 

127.9 

4.025 

21.25 

17 

45.26 

2,048 

6.200 

32.74 

161.3 

5.075 

26.80 

18 

40.30 

1,624 

4.917 

25.96 

203.4 

6.399 

33.79 

19 

35.89 

1,288 

3.899 

20.59 

256.5 

8.070 

42.61 

20 

31.96 

1,022 

3.092 

16.33 

323.4 

10.18 

53.75 

APPENDIX  B  427 

Stranded  Copper  Wire  —  100%  Matthiessen's  Standard 


No. 
B.  &S. 

Diam. 

Mils 

Area 
Cir.  Mils 

Weight,  Pounds 

Resistance,  20°  C. 
68°  F. 

Bare 

1000' 

Mile 

Feet 
per  Ib. 

1000' 

Mile 

2,000,000 

6,100 

32,210 

0.164 

0.005198 

0.02744 

1,500,000 

4,575 

24,160 

0.219 

0.006930 

0.03659 

1,250,000 

3,813 

20,130 

0  262 

0.008316 

0.04391 

1,152 

1,000,000 

3,050 

16,100 

0.328 

0  01040 

0.05489 

1,125 

990,000 

2,898 

15,300 

0.345 

0.01094 

0.05778 

1,092 

900,000 

2,745 

14,490 

0.364 

0.01155 

0.06099 

1,062 

850,000 

2,593 

13,690 

0.385 

0.01223 

0.06458 

1,035 

800,000 

2,440 

12,880 

0.409 

0.01299 

0.06861 

959 

750,000 

2,288 

12,080 

0.437 

0.01386 

0.07318 

963 

700,000 

2,135 

11,270 

0.468 

0.01485 

0.07841 

927 

650,000 

1,983 

10,470 

0.504 

0  01599 

0  .  08444 

891 

600  000 

1,830 

9,662 

0.546 

0.01732 

0.09145 

855 

550,000 

1,678 

8,857 

0.596 

0.01890 

0.09980 

819 

500  000 

1,525 

8,052 

0.655 

0.02079 

0.1098 

770 

450',000 

1,373 

7,247 

0.728 

0.02310 

0  .  1220 

728 

400,000 

1,220 

6,442 

0.819 

0.02599 

0.1372 

679 

350,000 

1,068 

5636 

0.936 

0.02970 

0.1568 

630 

300,000 

915 

4,831 

1.093 

0.03465 

0.1830 

590 

250,000 

762 

4,026 

1.312 

0.04158 

0.2196 

0000 

530 

211,600 

645 

3,405 

1.550 

0.04913 

0.2594 

000 

470 

167,800 

513 

2,709 

1.949 

0.06195 

0.3271 

00 

420 

133,100 

406 

2,144 

2.463 

0.07811 

0.4124 

0 

375 

105,500 

322 

1,700 

3.106 

0.09850 

0.5210 

1 

330 

83,690 

255 

1,347 

3.941 

0.1242 

0.6557 

2 

291 

66,370 

203 

1,072 

4.926 

0.1566 

0.8270 

3 

261 

52,630 

160 

845 

6.250 

0.1975 

1.043 

4 

231 

41,470 

127 

671 

7.874 

0.2490 

1.315 

This  table  is  calculated  for  untwisted  strands ;  if  the  strand  is  twisted  the 
cross  section  of  the  copper  at  right  angles  to  the  length  of  the  strand,  the 
weight  per  unit  length  and  the  resistance  per  unit  length  will  each  increase 
from  1  to  3  per  cent,  and  the  length  per  unit  weight  will  decrease  from  1  to  3 
per  cent,  depending  on  the  number  of  twists  per  unit  length  and  the  number 
of  wires  in  the  strand. 


INDEX 

Summaries  of  important  definitions  and  principles  will  be  found  on  the 
following  pages: 

Magnestism     ...........  93 

Continuous  Electric  Currents     ......  174 

Electromagnetism    ........  230 

Electrostatics            .                    .          .          .          .          .          .          .          .  276 

Variable  Currents     .          .          .          .          .          .          .          .          .          .  299 

Alternating  Currents         .          . '        .          .          .          .          .          .          .  360 

Symbolic  Method     ..........  391 

Three-Phase  Alternating  Currents       .          .          .  416 


Abbreviations,  422. 
Absolute  system  of  units,  13. 

See  under  name  of   quantity 

individual  units. 
Absorption,  electric,  273. 
Acceleration, 

definition  and  units  of,  10. 

of  rigid  body,  20. 
Admittance, 

definition  of,  349. 

of  an  inductance  and  capacity 
parallel,  351. 

symbolic  notation  for,  380. 

units  of,  351. 

Admittances  in  parallel,  351. 
A.  I.  E.  E.  style,  422. 
Alternating  current, 

average  value,  309. 

definition  of,  306. 

effective  value,  311. 

establishment  of,  353. 

instantaneous  value,  309. 
Alternating  currents,  304. 

addition  of,  331. 

polyphase,  396. 

use  of,  312. 

Alternations,  definition  of,  307. 
Alternator,  304,  396. 
American  wire  gauge,  425. 


Ammeters,  122. 
Ampere, 
for          definition  of,  111. 

international,  126. 
Ampere-hour,  128. 
Ampere-turns, 

calculation  of,  210. 

definition  of,  207. 
Amplitude  factor,  328. 
Analysis  of  wave,  320,  fif. 
in      Angle,  3. 

Anode,  definition  of,  125. 
Apparent  power,  330. 
Arc,  electric,  217. 
Armature, 

electromotive  force,  202,  204. 

of  a  generator,  198. 
Average  value,  309. 

B.  &  S.  wire  gauge,  425. 
B-H.  curves,  86. 

experimental  determination  of,  195. 
Balance,  current,  122. 
Balanced  system,  definition  of,  400. 
Ballistic  galvanometer,  194. 
Battery,  electric,  155. 


C.  G.  S.  system  of  units,  13. 
Cable,  insulation  resistance,  169. 


429 


430 


INDEX 


Calibration  of  meters.  122,  145,  194. 
Capacity, 

and    inductance    (see    Inductance 
and  Capacity). 

definition  and  units  of,  267. 

of  various  forms  of  condensers,  269 . 

reactance,  342. 

specific  inductance,  272. 
Capacities, 

in  parallel,  273. 

in  series,  272. 

Cathode,  definition  of,  125. 
Center  of  mass,  definition  of,  14. 
Charging  by  contact  and  by  induc- 
tion, 239. 

Charging    current    (see    Displace- 
ment Current). 
Charge,  electrostatic, 

definition  of,  237. 

properties  of,  242. 

and  quantity  of  electricity,  259. 

residual,  273. 

units  of,  243,  261. 

within  a  conductor,  250. 
Charge,  magnetic,  38. 
Chemical  energy,  23,  154. 
Circuit,  electric, 

definition  of,  102. 

energy  associated  with,  284. 

general   equations    of    the    simple, 

285. 

Circular  mil,  133. 
Coil    current    in    a    delta-connected 

generator,  402. 
Coil  electromotive   force   of   Y-con- 

nected  generator,  399. 
Commutation,  200. 
Complex  number, 

admittance  as  a,  380. 

division  of  a  rotating  vector  by, 
381. 

impedance  as  a,  378. 

multiplication  of  a  rotating  vector 

by,  378. 
Components  of  a  vector, 

power  and  reactive,  328. 
Compound  wound  dynamo,  202. 
Concentrated  winding,  120,  225. 


Condenser,  electric, 

capacity  of,  267,  269. 

charging  through  a  resistance,  292. 

co-axial  cylinders,  270. 

definition  of,  268. 

discharging  through  an  inductance, 
295. 

discharging  through   a   resistance, 
294. 

discharging    through    a   resistance 
and  inductance,  357. 

harmonic  current  through,  342. 

in  parallel  and  in  series,  272. 

non-harmonic  current  through,  343. 

parallel  cylinders,  271. 

parallel  plate,  269. 

reactance  of,  342. 

spherical,  269. 
Condensers  in  series  and  in  parallel, 

272. 
Conductance, 

definition  of,  136. 

effective,  349. 

units  of,   136. 
Conductivity,  136. 

Matthiessen's  standard  of,  136. 
Conductor,  definition  of,  103. 
Conductors,  electric,  103. 

in  parallel,  112. 

in  series,  112. 

Contacts,  law  of  successive, 
Continuous  current,  establishment  of, 

288. 
Copper, 

specific  resistance  of,  135. 

temperature  coefficient,  137. 

wires,  425. 
Corona,  266. 

Coulomb,  definition  of,  128. 
Couple,  definition  of,  21. 
Current,  electric, 

absolute  measurement  of,  120. 

alternating  (see    Alternating    Cur- 
rent). 

comparison  of  strengths  of,  122. 

continuous,  definition  of,  108. 

decay  of,  291. 

definition  of,  101. 


INDEX 


431 


Current,  electric — Continued. 
direction  of,  111. 
direction  of,  lines  of  force  produced 

by,  115. 

displacement,  263. 
establishment  of,  288,  353. 
force  produced  by  magnetic  field 

on  wire  carrying  an,  113. 
magnetic  field  produced  by,    115, 

ff,  170. 

oscillating,  307. 
pulsating,  307. 
stream  lines  of,  165. 
strength  of,  108. 
units  of,  111. 
variable,  284. 
Currents,  electric,  polyphase,  396. 

Delta-connected  generator, 

equivalent  Y  impedance  of,  405. 

relation  between  coil  current  and 

line  current,  404. 
Delta-connected  load,  400. 
Delta  connection,  397. 
Density,  definition  of,  13. 
Diamagnetic  substances,  37. 
Dielectric  constant,  248,  272. 

hysteresis,  275. 

strength,  266. 

Dielectric,  definition  of,  103. 
Difference    in    phase,    definition    of, 

308,  375. 

Discharge  from  points,  266. 
Displacement  current,  263. 
Displacement,  electric,  247. 

mechanical,  4. 
Dynamo, 

alternating  current   (see  generator 
alternating  current). 

continuous  current,  197. 

electromotive  force  of,  204. 
Dyne,  definition  of,  18. 

Earth's  magnetic  field,  49. 
Eddy  currents,  288. 
Effective  value, 

definition  of,  311. 

of  a  harmonic  current  or  p.  d.,  311 


Effective  value — Continued. 

of  a  non-harmonic  current  or  p.  d., 

319. 

Effective  resistance,  337. 
Efficiency,  definition  of,  31. 
Electrical  degrees,  396. 
Electricity, 

analogous    to    a    non-compressible 

fluid,  104. 
quantity  of, 

definition  and  units  of,  127. 
measurement  of,  193. 
Electrisation, 
intensity  of,  245. 
lines  of,  245. 

Electrochemical  equivalent,  126. 
Electrodes,  125. 
Electrolysis,  124. 
Electrolytes,  124. 

Electromagnet,  tractive  force  of,  229. 
Electromagnetic    induction,    electro- 
motive force  due  to,  185. 
Electrometer,  parallel  plate,  256. 
Electromotive  force, 

and    potential   drop,    relation   be* 
>       tween,  149. 
armature,  202. 
contact,  10. 
definition  of,  148. 
due  to  electromagnetic    induction 

185. 

impressed,  151. 
of    alternating    current    generator, 

305. 

of  chemical  battery,  102,  155. 
of    continuous    current    generator, 

204. 

of  three-phase  generator,  397. 
terminal,  151. 
thermal,  157. 
Electromotive  forces, 

externally  induced,  338,  350. 
in  parallel,  160. 
in  series,  160. 

Electrostatic  field,  energy  of,  274. 
Electrostatic  field  of  force,  244. 
Electrostatic      flux      density,      247, 
251. 


432 


INDEX 


Electrostatic  induction, 

flux  of,  247. 

b'nes  of,  247. 

phenomenon  of,  239. 
Electrostatic  intensity,  244. 

inside  a  conductor,  249. 

just  outside  a  conductor,  250. 
Electrostatic  potential,  253. 

and    electrical    potential,    relation 

between,  261. 
Electrostatic  screen,  249. 
Electrostatics,  237. 
Energy, 

associated  with  a  circuit,  284. 

chemical,  23,  154. 

conservation  of,  23. 
.  definition  of,  21. 

electric,  definition  of,  141. 

electric,  measurement  of,  147. 

electrostatic,  274. 

heat,  30,  228. 

magnetic,  216,  ff,  226. 

units  of,  24. 
Equipotential  surfaces, 

electric,  167. 

electrostatic,  255. 

magnetic,  92. 

Equivalent  sine-wave  p.  d.  and  cur- 
rent, 319. 

Equivalent  F,  reduction  of  all  bal- 
anced   three-phase    circuits    to, 
406. 
Equivalent  Y-connected, 

generator,  405. 

load,  405. 

Erg,  definition  of,  24. 
Externally  induced  e.m.f.'s,  338,  350. 

Farad,  267. 

Faraday's  Law  of  Electrolysis,  125. 

Field  of  electrostatic  force, 

definition  of,  244. 

energy  of,  274. 

intensity     of     (see     Intensity     of 

electrostatic  field). 
Field  of  magnetic  force, 

definition  of,  43. 

energy  of,  216  ff,  226. 


Field  of  magnetic  force — Continued. 

intensity  of  (see  Intensity  of  mag- 
netic field). 

Fisher-Hinnen  method  of  wave  anal- 
ysis, 322. 
Flux, 

of  electrostatic  force,  244. 

of  electrostatic  induction,  244. 

of  electrisation,  245. 

of  magnetic  force,  53,  57,  67. 

of  magnetic  induction,  67,  72. 

of  magnetisation,  66. 
Flux  density,  magnetic, 

and  intensity  of  magnetisation,  71. 

definition  of,  71. 

on  the  two  sides  of  a  surface,  73. 

units  of,  72. 

Flux  density,  electrostatic,  247,  251. 
Force, 

definition  of,  16. 

due  to  electric  charges,  238,  242. 

on  a  magnetic  pole,  due  to  electric 
current,  114. 

due  to  magnetic  field  on  wire  carry- 
ing a  current,  113. 

required  to  separate  two  magnetic 
poles,  48. 

units  of,  19. 
Force,  flux  of, 

electrostatic,  244. 

magnetic,  53,  57,  67. 
Force,  lines  of, 

electrostatic,  244. 

magnetic,  53,  57,  67. 
Force,  moment  of,  definition  of,  19. 
Form  factor,  328. 

Foucault    currents    (see    Eddy    cur- 
rents). 

Fourier's  Series,  305. 
Free  period  of  a  circuit,  296,  345. 
Frequency,  definition  of,  307. 

Galvanometer, 

ballistic,  194. 

tangent,  122. 
Galvanometers,  122. 
Gauss,  definition  of,  71. 
Gauss's  theorem,  56. 


INDEX 


433 


Generator,  alternating  current, 

single-phase,  304. 

three-phase,  396. 

coil  current,  404. 

coil  e.  m.  f.,  399. 

equivalent    Y-connected,   405. 
Generator,  continuous  cuurrent,  197. 

e.  m.  f.  of,  204. 
Geometrical  addition,  7. 
Gilbert,  definition  of,  92. 
Gradient,  potential,  92,  167. 
Gravitation,  Newton's  law  of,  32. 
Gravity, 

acceleration  of,  11. 

center  of,  14. 

specific,  13. 

Harmonic  current,  establishment  of, 

353. 

Harmonic  e.  m.  f.  and  current,  305. 
Harmonic  function, 

definition  of,  28. 

represented  by  a  rotating  vector, 
332. 

symbolic  expression  for,  376. 

symbolic  expression  for  derivative 

of,  377. 

Harmonic  functions,  addition  of,  331. 
Harmonic  motion,  25. 
Heat  energy, 

definition  and  units  of,  30. 
Henry,  212. 
Hysteresis, 

dielectric,  275. 

energy  loss  due  to,  228. 

magnetic,  82. 

Hysteresis  loop,  experimental  deter- 
mination of,  195. 

Impedance, 
definition  of,  337. 
symbolic  notation  for,  378. 
units  of,  340. 
Impedance  coil,  340. 
reactance  of,  to    o  harmonic    cur- 
rent, 340. 

reactance   of,    to   a   non-harmonic 
current,  340. 


Impedance  of  a  resistance,   induct- 
ance and  capacity  in  series,  344. 
Impedances,  equivalent  Y  and  delta, 
405,  407. 

in  parallel,  348,  383,  389. 

in  series,  346,  387. 
Impressed  e.  m.  f.,  151. 
Inductance  and  capacity, 

and  linkages,  213. 

calculation  of,  222. 

discharge  of  through  a  resistance, 
291. 

in  parallel,  351. 

in  series,  295. 

of  concentrated  winding,  225. 

of  a  solenoid,  225. 

of  two  parallel  wires,  223. 

self  and  mutual,  211. 

units,  of  212. 

Inductance   and   resistance    (see   re- 
sistance and  inductance). 
Induction, 

electromagnetic,  185. 

electrostatic,  247. 

magnetic,  67,  76. 
Induction,  flux  of,  39. 

electrostatic,  244. 

magnetic,  67. 
Induction,  lines  of, 

determination  of,  linking  a  circuit, 
193. 

electrostatic,  244. 

magnetic,   67. 
Instantaneous  values,  309. 
Insulation  resistance  of  a  cable,  .169. 
Insulator,  103. 

Insulators,  definition  of,  103. 
Intensity,  electric,  167,  263. 
Intensity  of  electrisation,  246. 

and  electrostatic  flux  density,  247. 
Intensity   of    electrostatic   field, 

and.jeleetrostatic  flux  density,  247. 

definition  of,  244. 

insjda_a~elesed-conchictor,  249. 

just   outside   a   closed    conductor, 

250: 

Intensity  of  magnetic  field, 
and  flux  density,  71,  76. 


434 


INDEX 


Intensity .  of.  -magnetic  field — Con. 

definition  of,  43. 

due  to  bar  magnet,  45. 

due  to  current  in  circular  coil,  118. 

due   to    current   in   long   straight 
wire,  116. 

due  to  current  in  solenoid,  191. 

due  to  point-pole,  45,  80. 

due  to  magnetically  charged  disc, 
47. 

in  closed  iron  ring,  196,  207. 

measurement  of,  50. 

tangential  components  of,  74. 

units  of,  44. 
Intensity  of  magnetisation, 

and  magnetic  flux  density,  73,  77. 

and  pole  strength  per  unit  area,  65. 

definition  of,  62. 

measurement  of,  66. 
Inverse    points    with    respect    to    a 
circle,  170. 

"j,"   meaning  of  symbol,  371,  374, 

378,  377. 
Joules's  Law,  129. 


Kilowatt,  definition  of,  146. 
Kilowatt-hour,  definition  of,  146. 
Kinetic  energy  and  magnetic  energy, 

analogy  between,  218. 
Kirchhoff's  Laws, 

for  direct  current  circuits,  158. 

for  the  magnetic  circuit,  209. 

in  symbolic  notation,  383. 

Lag,  definition  of,  309. 
Lead,  definition  of,  309. 
Leakance,  definition  of,  285. 
Left-hand  rule,  111. 
Length, 

equivalent,  of  a  magnet,  49. 

units  of,  1. 
Line  current,  408. 

voltage,  408. 

Lines  of  electrisation,  245. 
Lines  of  electrostatic  force, 

and  electrostatic  intensity,  244. 


Lines  of  electrostatic  force — Con. 

and  lines  of  induction,  244. 

due  to  an  electric  charge,  244. 
Lines  of  magnetic  force, 

and  field  intensity,  53,  57,  61. 

and  lines  of  induction,  67. 

definition  of,  54,  57. 

due  to  an  electric  current,  115. 

due  to  single  pole,  53. 

resultant,  57. 
Lines  of  magnetic  induction,  67. 

cutting  of,  188. 

due  to  an  electric  current,  116. 

measurement  of,  193. 

refraction  of,  78. 
Lines  of  magnetisation,  65,  67. 
Linkages,  definition  of,  185. 
Load,   equivalent  Y-connected,  406, 
Losses,  definition  of,  31. 

Magnet, 

definition  of,  37. 

equality  of  the  poles  of,  49. 

equilibrium  position  of,  50. 

equivalent  length  of,  49. 

frequency  of  vibration  of,  51. 
Magnetic  charge,  definition  of,  38. 
Magnetic  energy,  216  ff,  226. 
Magnetic  field  intensity, 

definition  of,  43. 

due  to  an  electric  current,  113,  ff, 
170. 

due  to  bar  magnet,  45. 

due  to  magnetically  charged  disc. 
47. 

due  to  the  earth,  49,  50. 

effect  of  surrounding  medium  on, 
40,  80. 

measurement  of,  50. 

tangential  components  of,  74. 

units  of,  44. 
Magnetic  field  of  force, 

definition  of,  43. 

energy  of,  226. 

Magnetic  flux  'see  Flux  of  magnetic 
force,  Flux  of  magnetic  induc- 
tion, Flux  cf  magnetisation). 
Magnetic  induction,  39,  67,  76,  79. 


INDEX 


435 


Magnetic  moment, 

definition  of,  49. 

measurement  of,  52. 
Magnetic    point-pole,    definition    of, 

41. 
Magnetic  poles, 

attraction  and  repulsion  of,  38. 

definition  of,  37,  38. 

equality  of,  42,  50. 

force  of  repulsion  between,  38,  41, 
80. 

force    required    to    separate    two 
equal  and  opposite,  48. 

properties  of,  41. 
Magnetisation, 

curves  of,  87. 

flux  of,  66. 

induced,  39,  76. 

intensity  of,  62. 

lines  of,  65. 
Magnetomotive  force, 

definition  of,  205. 

units  of,  207. 
Mass, 

center  of,  14. 

conservation  of,  16. 

definition  and  units  of,  12. 
Matthiessen's  Standard  of  Conduc- 
tivity, 136. 

Maximum  values,  309. 
Medium,  effect  of, 

on  electrostatic  forces, 

on  magnetic  forces,  40,  42,  80. 
Mesh  connection,  397. 
Meter-gram,  124. 
Mho,  136. 

Moment  of  force,  19. 
Moment  of  inertia,  16. 
Moment,  magnetic,  49. 
Momentum, 

conservation  of,  16. 

definition  of,  15. 
Motion, 

harmonic,  25. 

of  a  system  of  particles,  20. 
Motor, 

continuous  current,  197. 

electromotive  force  of,  204. 


Mutual  inductance, 

definition  and  units  of,  212. 

Natural  period  of  a  circuit,  296,  345. 
Neutral  point,  397. 
Newton's  Law  of  Gravitation,  32. 
Normal  components,  73,  253. 

Oersted,  208. 
Ohm,  129. 
Ohm's  Law,  146. 

generalized,  150. 
Opposition,  definition  of,  309. 

Parallel  arrangement, 

of  condensers  273. 

of  conductors  112,  160,  348. 
Paramagnetic  substances,  37. 
Pendulum,  motion  of,  25. 
Period,  definition  of,  307. 
Periodicity,  definition  of,  307. 
Permeability,   magnetic,   77. 
Phase, 

definition  of,  307. 

difference  in,  308,  375. 
Point-charge,  definition  of,  242. 
Point-pole,  definition  of,  41. 
Points,  discharge  from,  266. 
Polarisation, 

electrostatic,  247. 

of  a  battery,  156. 
Pole  strength, 

definition  of,  42. 

per  unit  area,  43,  79. 

unit  of,  42. 
Poles, 

magnetic    (see   magnetic   poles). 

of  a  battery,  102,  155. 
Polyphase  alternating  currents,  396. 
Potential, 

electrostatic,  253,  261. 

magnetic,  88. 
Potential    difference     (see    potential 

drop). 
Potential  drop,  electric, 

and  electromotive  force,  149. 

and    electrostatic    potential    drop, 
261. 


436 


INDEX 


Potential  drop,   electric — Continued. 
definition  of,  139. 
measurement  of,  134. 
units  of,  143. 

Potential  drop,  electrostatic, 
definition  of,  254. 
measurement  of,  256. 
units  of,  143,  262. 
Potential  drop,  magnetic, 

and  magnetomotive  force,  205. 
definition  and  units  of,  91. 
Potentiometer,  163. 
Power,  apparent,  330. 
average,  314,  ff. 
definition  of,  24. 

electric,  definition  and  units  of,  146. 
electric,  measurement  of,  147. 
in  a  balanced  three-phase  system, 

413. 

in  symbolic  notation,  385. 
in  a  three-phase  circuit,  measure- 
ment of,  410. 
of  a  harmonic  p.  d.  and  a  harmonic 

current,  314. 

of  a  non-harmonic  p.  d.  and  a  non- 
harmonic  current,  317. 
reactive,  330. 
units  of,  25. 
Power  component,  328. 
Power-factor, 
definition  of,  314. 
in  symbolic  notation,  387. 
in  a  three-phase  system,  407,  415. 

Quadrature,  definition  of,  309. 
Quantity  of  electricity, 

and  electric  charge,  259. 

definition  of,  127. 

measurement  of,  193. 

units  of,  127. 

Rating  of  three-phase  apparatus,  410. 
Reactance, 

capacity,  342. 

definition  of,  337. 

of  a  coil,  340,  341. 

of  a  condenser,  342. 

units  of,  340. 


Reactive  component,  328. 
Reactive  power,  330. 
Reluctance,  magnetic,  207. 
Reman  ent    or    residual    magnetism, 

83,  202. 

Residual  charge,  273. 
Resistance,  electric, 

absolute  measurement  of,  130. 

and  conductance,  effective,  351. 

definition  of,  129. 

effective,  337. 

inductance  and  capacity  in  series, 
344,  357. 

insulation,  of  a  cable,  169. 

specific,  132. 

temperature  coefficient  of,  137. 

units  of,  129. 
Resistances, 

comparison  of,  132,  162. 

in  parallel,  160. 

in  series,  160. 
Resistance  and  capacity  in  series, 

decay  of  current  in,  294. 

establishment  of  current  in,  292. 
Resistance  and  inductance  in  series, 

alternating  current  in,  335. 

decay  of  current  in,  291 . 

establishment  of  current  in,  288. 

impedance  of,  340. 
Resistivity,  132. 
Resonance,  295,  345,  351. 
Right-hand  rule,  188. 
Right-handed  screw  law,  188. 
Root — mean — square,  311. 


Saturation,  magnetic,  86. 
Screen,  electrostatic,  249. 
Self-inductance, 

definition  and  units  of,  212. 
Series  arrangement, 

of  condensers,  272. 

of  conductors,  112,  160,  346. 
Series  dynamo,  202. 
Shunt  dynamo,  202. 
Silver  voltameter,  127. 
Sine-wave    (see  harmonic   function). 
Skin  effect,  222. 


INDEX 


437 


Solenoid,  intensity  of  magnetic  field 
inside,  191. 

inductance  of,  225. 
Spark,  electric,  217. 
Specific  gravity,  13. 

inductive  capacity,  272. 

resistance,  132. 

Speed,  definition  and  units  of,  10. 
Star  connection,  397. 
Strength,  dielectric,  266. 
Successive  contacts,  law  of,  154. 
Surface,  units  of,  2. 
Surface  conditions, 

in  an  electrostatic  field,  253. 

in  a  magnetic  field,  75. 
Susceptance, 

definition  of,  349. 

units  of,  351. 
Symbolic  expression, 

rationalization  of,  382. 
Symbolic  method,  371. 

examples  of,  387. 

Symbolic  notation,  meaning  of  sym- 
bol "j"  in,  371,374,  377,378. 

Tangent  galvanometer,  122. 

Tangential  components,  74,  253. 

Temperature,  definition  and  units  of, 
29. 

Temperature    coefficient    of    electric 
resistance,  137. 

Terminal  electromotive  force, 
definition  of,  151. 
of  a  three-phase  generator,  399. 

Three-phase    alternating    currents, 
396. 

Time,  units  of,  3. 

Torque, 
and  power,  25. 
definition  and  units  of,  19. 

Transformer,      alternating      current, 
313. 

Transient  effects  produced  by  a  har- 
monic e.  m.  f.,  353. 

Transmission  line,  297,  409. 

Turning  moment  (see  Torque). 

Two  wattmeter  method  of  measuring 
power,  410. 


Units, 

absolute  system  of,  13. 

for  units  of  various  quantities  (see 
under  name  of  quantity  meas- 
ured). 

Variable  currents,  284. 
Vector, 

addition  of,  7,  372. 

components  of,  6. 

definition  of,  4. 

derivative  of  a  rotating,  377. 

division  of,  by  a  complex  number, 
381. 

multiplication  of,  by  a  complex 
number,  378. 

referred  to  another  vector  as  line 
of  reference,  375. 

representation  of  a  harmonic  func- 
tion, 332,  ff. 

rotating,  332. 

subtraction  of,  7,  373. 

symbolic      expression      for,      203, 

375. 
Vectors, 

composition  of,  6, 

definition  of,  4. 

difference  in  phase  between, 
Velocity,  definition  and  units  of,  9. 
Volt, 

definition  of,  143,  157. 

international,  157. 
Voltameter,  127. 
Voltmeter,  144. 
Volume,  units  of,  2. 

Watt, 

definition  of,  24. 
Wattless    component    (see    Reactive 

component). 
Wattmeter,  147. 

measurement  of  three-phase  power, 

410. 

Wave  analysis,  320,  ff. 
Wheatstone  Bridge,  162. 
Wire  as  a  geometrical  line,  108. 
Wire  tables,  425. 


438  INDEX 

Work,  Work — Continued. 

current    and    magnetic    flux,    re-  required     to     separate     magnetic 

lation  between,  187.  poles,  88. 

definition  of,  21.  units  of,  24. 
relation  between  current,  flux  and, 

187. 

required    to    establish    a    current,  Y-connection,  397. 

215,  ff.  reduction    of   all   balanced    three- 
required      to      separate      electric  phase  circuits,  2, 406. 

charges,  254.  terminal  e.  m.  f.  of,  399. 


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